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CLOSED GRAPH THEOREMS FOR LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES A Dissertation Submitted <-o the Faculty of Science, University of the Witwatersrand, Johannesburg in Partial Fulfilment of the Requirements for the Degree of Master of Science by JANET Johannesburg, 1978 MARGARET HELMSTEDT i ABSTHACT Let 4 be the class of pairs of loc ..My are such that every closed graph linear continuous. let & w It secondly when B * (B) B ,pp, 1 from X (X,V) into “h ‘ch V is is any class of locally . ivex l.ausdortf spaces. . (X . (X.Y) e 4 dissertation, whon B onvex spaces for ,11 Y E B). " ‘his expository is investigated, firstly i r arbitrary is the class of C,-complete B . paces and thirdly is a class of locally convex webbed s- .ces. ACKNOWLEDGEMENTS I would like to express my gratitude to my supervisor, Professor D.B. Sears, for his enthusiasm, encouragement and advice. 1 also wish to thank Mrs. L. Berger who typed the manu script so expertly. ill DECLARATION. I certify that this dissertation is my own unaided work ? neither the substance nor any part thereof has been nor will be submitted for a degree in any othei Uni.-rsity ; no information used ha, been obtained by me while employed by, or working under the aegis of, any person or organisation other tilan the University. Janet Margaret Helmstedt. CONTI'NTS ACKNOWLEDGEMENTS DECLARATION 1. INTRODUCTION 2. SOME GENERAL CLOSED GRAPH THEOREMS 3. A CLOSED GRAPH THEOREM FOP 4. A CLOSED GRAPH THEOREM FOR CONVEX SPACES WITH Q, WEBS 28 5. A COUNTER-EXAMPLE 45 6. SOME OTHER CLOSED GRAPH THEOREMS - A SUMMARY 51 AND NOTATION 1 -COMPLETE SPACES 6 19 APPENDIX 54 REFERENCES 60 INDEX OF DEFINITIONS 62 INDEX OF SYMBOLS 65 INTRODUCTION AND NOTATION 1.1 Introduction By "convex space" we meac "locally convex topological vector space over the field C of complex numbers". with the class-fcof pairs (X,Y) of convex spaces with the property that every closed graph linear mapping from uous. We are concerned X into Y One way to investigate ■O is to choose a class is contin B of convex Hausdorff spaces and then consider "6 (B) = {x : (X,Y) £ "6 Y € B}. Banach [1] showed in 1932 that if Frechet spaces, then (B) includes B for all is the class of B. In Chapter two we prove some general closed graph theorems which apply to every class B of convex Hausdorff spaces. lyahen in [7] we show that if spaces, then (i) B is any class of convex Hausdorff is closed under the formation of inductive limits and subspaces of finite codimension. show that if then 4 (B) I Following Following de Wilde in [5] we is an index set of cardinality d and C' £ 4 (B) is closed under the formation of topological products of cardinality d. mapping theorem: In this chapter, we also prove a general open if the class B of convex Hausdorff spaces is cl'sed under the format:on vf Hausdorff quotient spaces and if X £ Y & (B) onto and X Y £ P then every closed graph linear mapping from is open. In Chapter three, we show that if spaces, then ^ (B) B is the class of B^-complete is the class of barrelled spaces. The proofs of this chapter are adapted from the corresponding proofs for B -complete spaces in [15]. Following Robertson and Robertson [15], we show that every Frechet space is B-completo. We give an example of a barrelled space and of a B -complete space neither of which is a Frechet space, and so we have a generalisation of Banach1s closed graph theorem. Chapter j.our is concerned with the class of convex spaces with O webs. The theory of webbed spaces if due to de Wilde. He intro duced the theory in 1967 [3] and developed it in a series of papers, 2. the most important being [4). this theory: We quote Horvath (16). P ^) on "His theory is undoubtedly the most important contri bution to the study of locally convex spaces since the great drscoveries of Dieudonne, Schwartz and Grothendiecx in the late forties and early fifties". de Wilde's closed graph theorem (41 states that if „ is the class of convex spaces w i t h B webs, then 6 (B) includes the convex Baire spaces. The class B has good stability properties. It is closed under the formation of closed subspaces, continuous images, countable inductive limits, countable products and weaker convex topologies. duals. B includes the Fre'chet spaces and their strong Proofs of theorems about spaces with 6 webs are often very technical. Robertson and Robertson 115) have simplified many of these proofs by introducing what they call "strands" of a wch on a convex space. However, their proofs apply only to what they ca convex spaces with "completing webs", which are special eases of convex spaces with 6 webs. we have defined what we call t .c ™ , r H , with proofs similar to those of Robertson and Robertson. ,. We have also used this method to obtain Powell's result til), viz. if convex wausdorff space tu (X.t) has a 6 web. so has is the weakest ultr.-boinological topology for This -■ a most pleasing result. (X,T ) X TU TU stronger than Powell proves in addition that in any chain of convex Hausdorff topologies including weaker than whore 1 . all those are 6 webbed and none of those strictly stronger an have a ©• w e b . _ . in Chapter five we show that neither of the closed graph theorem., of C h a p t e r s two and three is a generalisation of the other. to do this we needed a barrelled space a e w e b such that (X.Y) t t . X 1» order and a convex space w Valdivia 118) provided us with just such an examp1e . In the above chapters, we also prove some "sequentta . V >' . graph theorems", where we derive the continuity of certain linear mappings from the assumption that their graphs are sequentially clos instead of closed. Chapter six concludes the dissertation with a short account. out proofs, of other closed graph theorems. ‘ ... « t am aware. Theorems 2.1 and 2.13 are new result, 1.2 Notation We give here a list of symbols and terminology which we use frequently and which are not standard. Other terminology will be introduced as needed in particular sections of this dissertation. Let A <A> <A> denotes the subspace of h(A) g(A) X X. spanned by A. h(A) denotes the absolutely convex hull of A. g(A) denotes the gauge of A (X,t ) be a subset of a vector space A in the case when is absolutely convex and absorbent. If we are concerned wi' t a sir. ;le vector topology I for a vector space X , we shall refer to the X , meaning the vector topological vector space space endowed with the topology t. We shall often be concerned with more than one topology for a vector space use the symbol X. (X,T) In such cases we shall for the vector space endowed with the topology Let Neighbourhood X and Y X t. be topological spaces. By "neighbourhood" in X , we mean neighbourhood of the origin. Topologies on If subspace specified, and M is a subspace of M X , unless otherwise shall be taken to be endowed quotient space with the topology it inherits from the topology on X , and and Y X/M with the quotient topology. Topologically X are said to be topologically isomorphic isomorphic if there exists a one-to-one, continuous, open spaces linear mapping T from X onto Y. T is called a topological isomorphism. a space Let a a topology If a property ot (X,T) is an Due 1 pair be has the pr ot space and The pad pair if X,Y) Y opological vector spaces, ) :y T ot , we say that i - an a (X,T) ^apology. of ve-tnr spaces is called a dual is a svbspuce of X* which separates the points of If X. Let (X,Y) % e x , y E Y , wc denote be a dual pair. y(x) by (x,y). (x,X y +A y ) = * J ' V J for a11 _ 1 ^ J \ X G C . we y ,y G Y and all L C ' We x G x i all Since regard each X x'e X as an element of separates the points of Y , and Y*. Then (Y,X> is a duel pair. MX,Y) topology Polar The weakest topology on X which makes each y G y continuous is called the weak topology on x determined by Yand denoted Let topology be adual pair and .9 a collection of (X,Y) W(X,Y) W(X,Y). bounded subsets of Y satisfying the conditions: (a) if A,b€«9 , there exists D ^ ^ with A U B C D ; (b) if A C.B (c) and D t(X,Y) s(X,Y) is a scalar then X topology for X this topology. subsets of of the elements of form called the polar topology on the We use the symbol ^ 0 If sets of Y , then Y , then of the dual pair consists ^ o is (X,Y) ^ - W(X,Y). 0 (X,Y) of allthe to denote consists of all the finite of all the absolutely convex limit of convex spaces , a base of neighbourhoods for a convex Hausdorff elements of ^ . Inductive XA € ^ -pans X. The polars in A" X W(Y,X) If compact sub- is called the Mackey topology and denoted K(Y,X) V X .Y) . bounded subsets of Y . can ed thestrong topology of the dual pair and denoted s(X,Y). We use the definition of Robertson and Robertson (I)51 p. 78 2) for an inductive limit of a collection of convex spaces. The convex space X is said to be the inductive limit of the convex spaces a F a under the linear mappings • xa is endowed with vhe spans X and X y oi u strongest convex topology which makes each continuous. X^ , X , if T 5. n x If X is a topological vector space for each i G I an index set 6) X. i G I 3 symbols i in I , then, unless otherwise stated, the X . represent tne product and H X i 6 i i G i direct the sum respectively of the X^ under the product and direct sum topoloaies. If X is either © X. , we regard each II Xi i 6 I i G I as a subspace of X , by identifying and X. of the spaces ix G x : p (x) = 0 the jth for all projection from j * i} , where X onto p^ is X^. When we write J , C- © C i. G i 3 t H C or © C, , each is i G I i G I always taken to be C , the complex field under the Euclidean topology. If L Z T is a mapping from a set is a subset of tion of X T and to Y X , then If continuous we shall say that f topologies respectively. 1,0 of T X,Y and t -0 X f the topologies on transpose is f : X -► Y continuous when Let Y , and denotes the restric are sets, and simply say that , the into a set Z. T-0 T* X Y a mapping, continuous if it is are endowed with the In some cases we may is continuous if it is clear what X and Y are. be topological vector spaces and a linear mapping. Let (x, T*(w)) = (T(x),w) T* : Y* -> X* for all x G x T i X -► Y be defined by and all w G Unless otherwise stated, we shall use the symbol for T*|y , , where Y‘ is the continuous dual of y *. T Y. 6. CHAPTER TWO SOME GENERAL CLOSED GRAPH THEOREMS Throughout this chapter, the symbol B denotes an arbitrary class of convex Hausdorff spaces. If f is a mapping from a set G , of f is the subset and are topological spaces Y ( (x,f(x)) the product topology on ially) closed graph mapping. Hausdorff and f :x £ x} and X * Y of X * Y. If , then f is said to be a (sequent It is easily verified that if (X,Y) X G is (sequentially) closed in is continuous then concern is with pairs Y , the graph , X into a set G is closed. Y is Our primary of convex spaces for which every closed graph linear mapping from X into Y is continuous. However, we shall prove some results for sequentially closed graph mappings when these are easily derived from or are similar to the corresponding results for closed graph mappings. analogous to those of & and let 4 We need, then, definitions & (B)given in be the class of pairs the introduction. (X,Y) of convex spaces with the property that every sequentially closed graph linear mapping from X all Y <S If V x into if continuous, and let 4 0,T and v are Hausdorff topologies for the set 0 and (X,a) -> (X,T) has soaces T : X -> Y is a a linear mapping, of the graph of T X remains topologies stronger than continuous duals closed when X and ),(Y,T )) C ■€> where (X,Y) C -fo of Y IfB (6 ) b , 2 T is a Y 01 and and Y are andY* , then have any other convex (X,d) ((X,□),(Y ,t)) < X and w(Y,Y') and (Y,l) then isany topology of the dual pair 1 is any convex topology for Y Another easily verified fact is that if and then(X,Z) E 4 C 1 0 X' the topologies W(X,X') are convex Hausdorff spaces with T. Y are vector then the graph of respectively.It is now easy to show that if weaker than and closed graph linear mapping, where stronger than X , with X x Y. From this we may concludethat if convex Hausdorff spaces with (X,X') for T , then the identity map a closed graph.If and T : X -» Y vector subspace ((X, 0 4 ] b }. weaker than both I : (B) = {x : (X,Y) E Z is a closed (sequentially closed) subspace ( & ^) » then 4 (B )3 4 (D ). 1 2 The aboveremarks show that 1( i, a proper subset of NOTE 2.1 =2 . it may happen that Closed graph theorems are examples of a certain kind. Let a a (B,) ' < ' a good source of countei be a property of convex spaces» and suppose that every specs is Hausdorff.I a class of convex spaces such that every a ^ i s space is then, in any chain of convex Hausdorff topologies for a vector space x there can be at most one two a For example, if T. By symmetry, <X,T) Frechet space if 0 0 and the identity 0 into V 0. (X,o) is not a distinct from T.A similar mapping shows that if and is stronger than is a Frechet space, then is weaker than x I, T is any convex Hausdorff topology for comparable with, but X For, if has a closed graoh and so it is continuous and is stronger than then topology. topologies of this chain, the identity mapping I • (x,0) - (X,T) 0 o CX.T) X consideration of & B and (X.o) x. are topological spaces and T a mapping from Y , then, in order to prove that the graph of T is (sequentially closed), it is sufficient to show that if net (sequence) in then X y - T(x). which is such that xx - x and closed <*x> '= * T(xx) ' y The following useful lemma is easily proved u.,ing this technique. LEMMA 2.2 X, Y. Z Let continuous th e be topological spacesand gra p h o 'Y ■* 2 of g . f let f : > - X be (e nave a : X ■> Z is (sequen We consider the property "Hausdorff" in connection with closed graph theorems. L . n where % u "-m, L Let and so ositionll*. L X be a topological vector space and let is the set of .11 neighbourhoods in is a closed subspace of The quotient space X/L X X(H). if x - X(H). „„ L Let It is immediate that is Hausdorff The topology on M X X topological direct sum of X onto M and and is called X . and we is Hausdorff if and only induces the indiscrete topology be ,n algebraic supplement of projection mapping from men (see 1151 P 26 Prop the Hausdorff topological rector space associated m t h denote it X. L L in X. in continuous and so b and M Then the X is the is totoiogical y '.The proof of this theorem given in 1151 does not require the hypothesis that the spaces concerned are convex. isomorphic to iae,;tlfy M X(H). and (See (15] p 95 Proposition 29.) X(H). We call the Havodorff e umand of M Suppose L and M X , then and if X L thei»dt8cr6t6 8 ^ « W a n d X. is a non-Hausdorff topological vector space. L * (O). If n e L , then into Let be respectively the indiscrete and Hausdorff summands of 1= * (xx) i" X. For each positive element of Y. Choose T (y.) •> n , but closed. * converges to also converges to be any other topological vector space and y We shall integer h € I. with T (0) # n , and x + n. T Now let x , Y a linear mapping from i ,let y. be the zero n * 0. Then so the graphof T > 0 and isnot sequentially Thus there are no closed or even sequentially closed graph linear mappings from a topological vector space into a non-Hausdorff topological vector space. or For this reason, when considering 6 (B) & (B) , we shall always assume that the spaces in B are i Hausdorff. Let us now consider non-Hausdorff domain spaces. Let X and be as in the previous paragraph and let x into Y. Let (xX) for each be a T be a linear mapping from net in x>- x ♦ n n G L. Suppose can only have that Thus the graph of a linear mapping from y = T(x + n) Y X which converges tox.Then T ( x x) for each converges to n G if l y. T(L) We = a non-Hausdorff topological vector space into another topological vector space can only be closed or sequentially closed if it maps the indiscrete summand of its domain into zero. zero mapping If X has the indiscrete topology then the is the only closed graph linear mapping from a topological vector space Y , and so (X,Y) • & graph theorem; hoZde for the pair Let X = M © L , where and indiscrete summands of X graph linear mapping from X projection mapping, and above remarks, f o l l o w s (X(H),Y). Suppose the closed graph theorem holds for the pair (X(H),Y). M y , since • of topoZopieaZ vector spaces if and onlb if it holds for the pair a net in into The closed graph theorem (sequentially closed THEOREM 2.3 Proof. X T and into Y. S : M + Y vanishes on T agree on that the graph of and L are the Hausdorff respectively. which converges to S M Let 1 m G m S = be the t |m . T - S " T. and if S(m^) and the graph of T Sis closed. be a closed : X> M be defined by L , and so M Let T if By the (m^ is converge? to is By hypothesis, S closed, it is 9. EEEH We note that the above proof goes through if X/L 1= any 3 = r r r r r : : r " r r . , nets where necessary. ##### m w m assumption without loss of tenerality. theobem 2.4. e,=h A <B> T . T a W A/B, ^ - is continuous snd so is ^ T (see (15, P Proposition 5). corollary.A (B) and *,(“ are closed under the ' ' .... . direct suns, quotients and finite products. I£ X - topology, ® Cj , then every^lineer X is equipped with the strongest convex napping £ ; n „„,.„uous and so X 6 A (=) X .etc any top.logical vector f°r 10 A subspace V codimewion if shown that of a vector space X/X 4 SB) X is said to be of A-'-.ic ha, finite dimension. lyah.n in [-1 has is closed under the formation of subspaces o finite codimension. Before we prove lyahen's result we prove a lemma which can be found in Kelley [91 p . M 5.5. LEMMA 2.5. If :%=y ' L = -*- U ::: r r be the closure of G in y) G 5 n (Z X Y) = closure of (0 T l t l t into Y. in Z xy (O.y, e = .then G. Hence .i.— Let V y : rTt of ’ X be a closed neighbourhood in o be an absolutely convex neighbourhood in m Since - r x x y . Let G linear2 ;— „ X > \J =* Ttx^) G V. a X Y. such that >:A P E ' L V is closed, T(x) V required result follows. theorem X 2 .6 . tef X he o of finite codimenoion. Proof. Y G B If T Thus " ' " % % p o s c X, * tubspa* of r'° u '‘o ’ is * hypcrplane in ba a closed graph linear mapping from closed graph linear map from in ^ G ^ (B) 3 Suppose firstly that and let thesis. cohdem apooe dtW X, X into Y T , the restriction of the domain of X X Y. Let T T_ . X - Y is X. Lo Xq and is continuous, by hypo Tj to XQ X^ . then is also T_ - T and be a linear extension of O is . 11. then the graph cf a € x \ X^. Since is closed,the graph of is = G + <(a,T? (a))> , where <(a,T va))> is or e dimensional and G also closed by Schaefer [16] p 22, 3 3. continuous and so is By hypothesis, ^ T. It is easily shown that the theorem is true when Xq codimension in (Z,Y) t is of finite X. In Chapter 3, we spaces with is shall give an example of apair Z (X,Y) € ^ , and a closed sub space (X,Y) of X of convex such that , thus showing that some limitation is needed on the codim ension of X in the above theorem. We come now to a closed graph theorem for products of convex Hausdorff spaces. This theorem was proved Ly de Wilde in a very interest ing paper [5]. We quote from the introduction to this paper. "The aim of this paper is to show that, in the study of products of topolog ical vector spaces, two kinds of subspaces play an essential roles the factor spaces and the simple subspaces which are the products of one-dimensional subspaces contained in the factor subspar- s. is a Hausdorff topological vector space for each then de Wilde's "simple" subspaces of II X^ isomorphic to We shall prove that 4 II C i 6 i . c 6 -6)(B) ( -fe (B)) , with cardinal I = d. i G I 1 topologically (B)) d X^ in an index set I, are each under the formation of products of cardinality H i If is closed provided that This is a particular 1 case of de Wilde's closed graph theorem for products. (His theorem is also valid for linear mappings whose graphs have other properties, e.g. the property of being a Borol set.) In this dissertation we are concerned primarily with convex space-,, but as de Wilde's Theorem is no more difficult to prove for topological vector spaces, we shall In this section on give the proof products of topological always denote the product ^ H ^ space for each i for this <ase. where in an index set X. I. vector spaces, X shall is a topological vector The following notation will be used in the next lemma and theorem. If then we If f € X , 1 € I and p. : X denote p^(f) f£* f 6 X and A C I , let f G X. is the 1th projection map, X be defined by (f^ = ^ if Then if f vanishes on A , f vanishes on I \ A , {A is a finite partition of = f , and in particular, f We shall regard each shall identify X. and X, l\ {i} = - where (and hence of X) it }s topologically isomorphic to he n sequence in simple subspace such that ^ ^ 0 m , let if f. ^ = 0 for all m ,(m) fi “ xiThen f (m) G xf for all that for each such that, for each Tnan there exists a f (m) e Xj, for all with U t% in m. xi ^ 0 and let A c I X such V t *21 such that V + V C U, Xj. oj X and for each U e U such that gA - 0 * g e U. there exists a finite set A C I and such that nA = 0 ~ y e u. Suppose the assertion is false for A is Hausdorff, be a family of balanced subsets of 9 ^ Xj, and each finite set X it is m. there exists a finite set g X G x^ Suppose that for each simple subspace Proof. and if x. f . = f ^ n) , V G ^ , there exists g £ X X ts non zero. choose Let % Then, for each If fi # 0 be defined as follows: for some THEOREM 2.8. f. ,( m ) f\ f G x > is the one-dimen- II c . i G I i € I , at most one of the Let < spanned by clear that each simple subspace is closed in Proof + f In particular, if called a atmpZc aubepace of ‘f lemma 2.7. Let (f(m ) then as being embedded in X , that is, we {f G x sional subspace of I I , there exists g G x U € with . q Then, for = 0 and $ U. Let V G be such that V + V C u. We construct by induction a sequence and a sequence (f(m)) in x (A^) of finite subsets of with the following properties: I , 13. (a) P. n An = 4> » for all (b) f (n) = for all m ^ n ; m ; m (c) t v Now for all U * X , so choose space containing h. h 6 x \ U. = B and and X^, f <n B < I (c) for h = f U) + hJ x R. 1 < k < m. Also M U. Let such that D C I f vanishes on ^ g. - D\ %nd Now A^. one"f|m) X£ such that f f , there exists a finite subset c)D ■ 0 and D is not meet each Choose ... 4 .W*. nn vanishes on This contradicts D i G I , there exists of k D. D. I lemma, there for each m. For such that ...................... (l) A are pair-wise disjoint, so D does m , ,(k) _(k) such that D n AR = <t>. Now - ^ Also Also f f ^ G c X,. X £* By (i) , f ^ G V. (c) , and the theorem is proved Let the product topology on COROLLARY. ^ ^ f0ll°WS that are satisfied bythe e Xf g G X f. -» g G finite and the A . m f (k) lK) and so There exists which is non-zero. By the previous exists a simple subspace Now and g - A (i and the f ^ . The Am are mutually disjoint, so, for each this \ f^ = 0 => f G V. - f M + gi\ D* AlS° g ^ U and gi\ D 6 V f (in) f v. 1 Hence conditions (a) , (b) and (c) most (a) , (b) such that f G X , and Let g Xc , be a simple subspace containing a finite set U m-1 There exists g G x at such that - 0 =» f e v. ^ Now = be a simple sub h ! \ D G v ' hence f 14 v Suppose that AR , f (k) have been defined to satisfy and g ^ Let There exists a finite set f G xh, and Let m. other rector topology or, X X he p. 1f any t which induces on each factor eubeapce and on each simple eubepace a topology weaker than that induced by p , then is weaker than r ?rooj_. on each Let <U be a base of balanced and let XE hood in X f. M v Ji 4 C- induce the topology X f , with similar notation for Let = T p. A ’ en u n Xf Hence there exists in X f a basic j * < f i> A , where A is X^ and Tf Let U p. X. n each T - neighbourhoods be a simple subspace of x ^ in X. ’s a p f "' ighbuur Pfneighbourhood a finitesubset of %, I , Vi isa neighbourhood in If g G xf <f/> and for gA = 0 i 6 then a and MC u n X f. Thus % g G M C u. satisfies the conditions of Theorem 2.8. Let V E % be such that exists a finite set Suppose B contains W + W +...+W C v Let hood in If B C i (m V + V C u. such that m = 0 , f f- X =* f € v. elements. summands). By the theorem, there Let W n xi W E % be such that is a p.-neighbourhood in X ^ . Q *= {f S x : fj ( W l') x^ Vi G B } , then Q is a p-neighbour- X. f e Q , then f = f x ' \ + f/-\e V+W+W+...+W C U (m+1 i G B summands) . Thus Q C u and T is weaker than p. We now deduce a remarkable theorem about linear maps acting on product spaces. Let THEOREM 2.9. map from X Y into be a topological vector space. Y is continuous if and only if its restrictions to the factor subepaces and to the simple subspaces of Proof. Let T be a linear map from X into restrictions to the factor and simple subspaces of Let be a base of balanced neighbourhoods for *0.' *■- fT 1 (U) : U G % } , then it 1 vector topology , O , on X.Let factor subspace of X , then neighbourhood in 2 induces on Z topology on A linear as are continuous. Y whose X Y. are continuous. Let it a base of neighbourhoods for a U G . if T"1 (U) Gi z = (Tj T ? X is continuous. z is a simple or (U) , which is a It follows that o a topology weaker than that induced by the product X. By the Corollary to Theorem 2.6, the product topology on X , and so each inthe i roduct topology and T T"1 (U) 0 is weaker than is a neighbourhood is continuous. da Wilde's closed graph theorem for products is an immediate consequence of this theorem. theorem 2.10. and so is Y Suppose that jbr each i Y is a topological vector space in an index set I. Suppose also that is Hausdorff and that the (sequentially) closed graph theorem holds for the pairs ( H C . , Y) i £ I and (X. , Y) 1 for each i G I. Then 15. tHe (ecquer.tially)closed Proof.r.«t „ OUSa r £f. Let Theorem 2.9, T x T SrapH H xv «= b A M n e e r « « t l y that each X i M ppin, from X into isomorphic to HOW X 's is continuous. Hausiorff in order that the simple subspaces of for the pair f. " ^ V in the above proof, we need the hypothesis that the riands'of tHcooer. N are topologically C i' xv l ( X X Then the (sequentially, closed graph theorem holds „ » 1 , T) by the above proof and Theorem 2.1. is topologically isomorphic to J / i * J , h ' " is easily proved that these factors are the Hausdorff and indiscr.tr summands of X. By Theorem 2.3 the (sequentially, closed graph theorem holds for the pair (X,Y). We restate the above theorem for convex spaces. m d srthe We note that C.et 4 f I." THEOREM 2.11. fomaUor 4 (B) 4(B) ofp Z Z A s of ,where is not always closed under the formation of 1* not aiwaya ^ iroducts, even if countable. Let X - ^ and let s , P be Then X 16 . spaces and examining the corresponding range spaces was used by Komura [10]. Powell [13] has expanded his result, and we give a short discussion of Powell's paper in Chapter 6. We shall now prove a lemma to bo found in Robertson and Robertson [IB] p 114 Lemma 5, and from it we shall dervve ^ useful open mapping theorem. 2.12. Let lemma a linear mapping and only if Xand Y of X intoY. (T')~' (X’) be convexHauedorff spaceo Then thegraph is dense in Y' of T and is closed under the topology T if W(Y’ , Y) Proof. X X* T" Y Y' Polars will be taken in graph of for X. T. Let % X, Y, X* or Y'. Let G be the be a base of absolutely convex neighbourhoods We shall first prove the following three results: (i) (( T T 1 (X-)) 0 = n TfTuTf ; U (ii) (T')~ 1 (X1) is dense in if and only if n (iii) y E Y' inder the topology W(Y' , Y) ((T')"1 (:(')) ° = {0} ; T(U) if and only if (0,y) G g. U eU (i) X* = U U° , hence U (V')-'(X') = U (T')"‘ (U°) = V €z% Thus ((T1)'* (X'))0 = (ii) (T')r ' (X') if and only if (iii) hood Let V in . Y' y E T(U) , there exists (T(U))C (T(U))00 = H U E % x E T(U). under the topology ((T')"1 (X'))°° = Y' U E Y O U E is dense in U U EM, W(Y* , Y) if and only if ((?')" (X'))e= ( Y 1)e={0> if and only if for every neighbour u with T(x) E y + v. Thus y E n T(U) if and only if for every neighbourhood V in Y and U E%, every U E , the neighbourhood (U, y + V) of (0,y) meets G. Thus y E n U E%/ Now suppose T(U) G if and only if is closed. (0,y) E g. By (iii) , (0,y) E g if and only 17. if y G TUir = ( ( T ' r 1 <X'))° O only i / y ^0. (T.)-‘ (x1) Hence is dense in Let Y* y - T(x) E X is dense in W(Y* Y' By (ii) , Y) . under the (x, T(x)) € 5 x y , (0,y - T(x)) G G. ((T')-'(X'))". (0,y) e G if and and so by (ii) , under the topology (x,y) E 5 , then vector subspace of But ((T1)”1 (X'XF = (o) Now suppose that (T')"'(X') topology. by (i). and since By (iii) , y " ?(%) W(Y* G , Y) is a and d) ^nd the graph of . G is closed. COROLLARY. Proof. Jf Let gropA o f ? M = (T')"1 (X') . 1 (0) « T-‘ (M°) = (T'(M))0 • <8 otoced, tAan From the lemma. Since (T'(M))° rVo, ia cZoczd. M° « {0>. is closed, so is T”1 (0) . We say that the open mapping theorem holds for the pair of convex spaces If every closed graph linea' mapping from Y Hence (x,.) X onto is open. Let THEOREM 2.13. md X V be ccmtcx Hauedoeff epaees. The following conditions are equivalent: (i) the open mapping theorem holds for the parr fit, c W a d gnspA tiheor a Hauedorff quotient cpaoe of Proof. mapping from Suppose (ii) is true. X onto Y. Let Let Let to- one linear mapping onto of ^ T Y. o"** be a closed graph linear Let X/M Z - X/M T - S . K , where Let * is continuous. M - ker T , then by the corollary to the previous lemma. the canonical mapping. X (Xt\) : is Hausdorff and S , Z + Y K' : z* -*■ x* K . X is a onc- be the transpose K. X X* Z=X/M Z* 1” I By L e m . 2.12. 1. d=n.. In ( V ,W(Y •,T)). .na so ( I T 1 ( X U M S T ' . I K ' ) - ( X ' I M S T 1 W ) t, follows that (S')"1 (%') is dense in Nowt'-X'-S'. (s«o Appendix II). (Y. , W (Y « , Y)) graph of S"‘ and by Lemma 2.9, the graph of is closed. By hypothesis, is an open mapping and since K S'1 S and hence the is continuous, has the same property, T S is an open mapping. Now suppose, (i) is true and let one linear mapping from of X . let Let K Y onto S be a closed graph one-to- X/M , where be the canonical mapping from M is a closed subspace X onto X/M and S'1 ° K. T Y X/M K Since 2.1, T X is continuous and S'1 has a closed graph, by Lemma has a closed graph and is open by hypothesis. Every open Since set in X/M is of the form K(U) where U is open in > is an T(U) is open and T(U) - S” 1 * K(U) , it follows that S' open mapping and so S is continuous. In practice, we use the following corollary to derive open mapping theorems from corresponding closed graph theorems. COROLLARY. U If B is a class of convex Hausdorff spaces which closed under the formation of Hausdorff quotient spaces, then the open mapping theorem holds for the pair ye. & (b ). (X, M if 1 ' and CHAPTER THREE A CLOSED GRAPH THEOREM FOR B^COMPLETE SPACES. A subset A of the continuous dual X' nearly .s said to be •very neighbourhood U in of a convex space A n u" i o l t ef X is said to be: closed for (i) (or fully comp-ete) if every nearly closed subspace of closed, « (x ' , x> is X. The convex Hausdorff space <(x. , x) X X' is Br-eonplete if every nearly closed (11) *(X' . X)-dense subspace of X' is W(X' , X) closed, and hence the whole of X'. In (15) p 107 corollary 2, it is shown that a convex Hausdorff space is complete if and only if every nearly closed hyperplane in is w(x’ , X) and every closed. Thus every B-complete space B -complete space is complete. X' is B^-complete We shall later give an example of a'complete space which is not Incomplete. To my Know ledge, it is not known if there exist B,-complete spaces which are not B-complete. It is easily verified that a B-complete (B-complete) remains B-complete (B-complete) space X under any stronger topology of the dual pair (X , %'). , A preliminary lemma is needed before we prove that every Frechot space is B-complete. LEMMA 3.1. in Let be U theconvex Haul ’orff a b so lu te ly convex ubset s Proof -A U u C A' ♦ U C A" + U. absolutely convex hull of . (A n , and sc To prove this, let that A" U „ rs y + u *r x + U • Then evace , X I'd of X'.(A ' Also, the cl- urn of the W U W _ J ' 7 (A <h U',' C A° + U. x 6 A' ♦ U a closed Now and choose A" + y <5 A- 1 I * and u « u , e y a u t U C A- t 2U. such The result now follows. THEOREM 3.2. Proof. spaceTf” X ’. Let Eifery Fvechet space is B-complete. X be a Frechet spaca and We show firstly that if neiahbourhood in X , then U M a nearly closed sub- is a closed absolutely (M ^ U°)° ' M" + 4U. 2U' Let n n since "u° , 3 Uj . and so 1 »Uh n u° = A . n A ^ l " “n .... ■ xn e 2Vn ana a _ n ^ By Lemma 3.1, :t.i•/.•» ................................. U ) 6 ^ r=l Now T r=n Hence to a point = :"n + ="n+l ' c ?.U + U + |Un + ...... { , x in - x0 * o e 4 U l" Le; y e ^ (M° + U)0 C seance X. »=« ^ , Then „ c , y h sincc^ • X, . for all sufficiently large u ^ ;t follows from h . . M s , . . . . h - • is nearly closed. X , an r ^TTLed, . . M n u; " n fttlV ’)B O » and since a subspace of - in C a u c h y 1 u c 2p -1 n ^ + u)o e {z € X' ; |(x,z)t < 1 T x e Z : . : tot y e Slnce U = » closed absolutely convex neighbourhoods for - - . — — there exists an ^ ^ ^ ^ 21. The following definitions and proofs lead up to the closed graph theorem of this section. Let X mapping. and Y be convex spaces and X for open if t W in Clearly, if T Y is said to be nearly continuous if T neighbourhood in X. T : X T'’ (V) every neighbourhood is a neighbourhood in T a linear Y V Y ,and nearly in for every neighbourhood is one-to-one and maps is nearly continuous if and only if is a T X onto U V , then is nearly open. It is easily verified that if Y is barrelled and T(X) m then nearly open and that ifX is barrelled , T is continuous. Let IX,T)be barrelled and let 0 which is strictly stronger than be a convex topology for T (for example, let infinite dimensional Banach space, and let as (X,0) I i (x,l) -+ (X,0) is continuous (X,T) o = t(X,X*) is not normable (see Appendix I). not open, for it is not nearly T is X be an 0 * 1 , , then The identity mapping and nearly open. However it ir. a topological isomorphism. The inverse of the above mapping provides an example of a mapping which is nearly continuous but not continuous. Let LEMMA 3.3. nearly continuous X and Y be convex Hausdorff spaces, linear mapping of X closed subspacs of X'. Proof- into Then f T T V W x X and M T a a nearly is nearly closed. x* T' Y Let is V be a W(Y’ and ,Y) neighbourhood in Y. closed. (T-1 (V))0 n (,r i (V))° n M T. is w(Y' W ( Y . , Y) Y' We must show that (r ) T"1 (V) is a neighbourhood in is W(X' , X) closed, since is also W(X* , X) closed (see Appendix m , Y) - W(X* closed. , X) Mow continuous, M X. (T (T’ (T')-‘KT-' (V))° n (M) ' (V)°~(T ' (V))° is nearly closed. III) and since ' (V))° ° M] ( T (T'’ (V))) ° ^ is (T -)’ ' (M) = (vnrtx))0 O (T1)"1(M). Also, ( T T ‘ (M) intersection of wry . y) W(Y' ,Y) closed sets. Hence which is an (T')"1 (M) ^ V is closed. theorem 3.4. T n v» « (T*)-’ (M) n v» n (V n T(x))° L e t X and Y be convex Hausdorff spaces and let be a nearly continuous closed graph linear mapping from X into X. If y is B^-oomplete then Proof. T We first show that continuous by showing that subspace of X' subspace of V. V. Y Since Let Then V V T is W(x , X') - W(Y , Y') T'(Y') C X '. and so by Lemma 3.3, x' is a nearly closed (T')" (X ') By Lemma 2.12, (T')-'(x') is Br-complete, is is a nearly closed w(Y' , Y)-dense in (T')_l (x') = y and so T'(y') C x'. be a closed absolutely convex neighbourhood in is also W(X , X') closed and hence closed. T is continuous. closed and so T' is nearly continuous, but T" 1 (V) is W(X , X') is a neighbourhood in T" 1 (V) - TT' (V) and so Y. X , as T is continuous. The closed graph theorem for ^-complete spaces is an immediate consequence of the above theorem. THEOREM 3.5. (Closed graph theorem for B^-complete spaces.) Zf J and f ore conwcz epoces uitA % AorraZZcd and y & -compZcte then (XjY) € 6 Proof. . X(H) r is barrelled (see [15] p 81 Proposition so by Theorem 2.3, we may suppose that be a closed graph linear mapping. neighbourhood in absorbent. Y. Since X and so T continuous. (v) X X Let is Hausdorff. V 6 ), and Let T « *>Y be an absolutely convex is closed, absolutely convex and is barrelled, T“ is nearly continuous. 1 (v) is a neighbourhood in By the previous theorem, T is -LARY. j ZyccmpZete apace c m m o t Aaw, a strictZy weaker barrelled Hausdorff topology. P£2o£'Wo use Note 2.1 to prove this result. We may now give an example of a complete convex space which is not Br-complete. Let (X.T) be an infinite dimensional Banach space. Under its strongest convex topology t (X , X*] Hausdorff and complete but not n o m o b l e T * T(X , X«l. By the above corollary , X is barrelled, (see Appendix 1 ). Thus (X , T(x , X*)) Is not -complete. We now show that Theorem 3.5 is a generalization in two senses of Banach's closed graph theorem. (see [15] p 61 Theorem 2 Every Frechet space is barrelled ) and B-completn (Theroem 3 .2 ). jn the previous paragraph, we gave an example of a convex Hausdo.-ff space which is barrelled but not normable. It remains to give an example of a B-complete space which is not a Frechet space. tt.h I uncountable. Frechet space. Then X is not metrisable and so The continuous dual of W. r its strongest convex topology, is , , and every subspace of every subspace of X' is Let X' W(X' X T(X' is X , X'*) % " _ " X is not ^ j C i‘ , the dual of is closed (see Appendix I). , X)-closed, and X X' Thus, is B-complete. We proceed now to show that every Hausdorff quotient space of a B-complete space is B-complete. This will enable us to derive an open mapping theorem for B-complete spaces. are needed. Note: if A is a neighbourhood in the convex Hausdcrff space then the polar of A Let LEMMA 3.6. T : X ■* Y Some preliminary results in X is the same as the polar of X' and Y Y'. Proof. The* n X*. M a near ly closed is nearZ* closed Polars will be taken in X in A be convex Hausdorff spaces , a nearly open linear mapping and ewbepoce of X , X ,X * , Y and Y'. X* T' Lct „ be a neighbourhood in note, and no T .(„) n uT .(«, OU- T* (M) n «- n u- - T M M ) n U - . is - W(x* T M H O . X) (T')" closed. Y , and so COS-.CC. « Since . :osed and compact. T . ( M n (TIU)) compact and closed. W(V (T (U))°) • TIW is a 1 3 W(Y ' ^ M " TIIU,,' rs W(y .X) . V. - W(X* . X) continuous and so °)is MIX* . x) Thus - U‘ by the above We shall show that (T(U))C - r ( U ) ) ° is nearly closed, T' is x' U" n It is easy to show that (U">> - T ' W r , neighbourhood in T'(M) n x' compact.Since the i s nearly closed in W(X* , X) topo- X. If there is d continuous nearly open mapping of a apace onto a conoer Hausdorff space I . then I -completi THEOREM 3.7. y-c^let, X , then Proof. Let T B-complete space be a continuous nearly open linear mapping of the X onto the convex Hausdorff space Y T' Y* Sine i •? uous. Let T'(M) is continuous, M T'(Y') C %' be a nearly closed subspace of is nearly closed in X'. weakly closed. Also, T(X) = Y dix IV). (T') Thus implies that and M is 1 Since X W(Y' , Y) Y'. is weakly contin By Lemma 3.6 , is B-complete, T'(M) , and so T' (T1 (M)) = M , and T’ is one-to-one is (see Appen theweak continuity of T1 closed. The quotient of a B-complete apace by a cloaed COROLLARY 1. aubapace is B-complete. Proof. Let M The canonical map be a closed subspace of the B-complete space K : X -*• X/M is open and continuous and so X. X/M is B-complete. COROLLARY 2. If a barrelled Hausdorff apace is the continuous image of a B-complete spacet it is B-complete. COROLLARY 3. X {Open mapping theorem for B-complete spaces.) is barrelled Hausdorff and theorem holds for the pair Proof. I If is B-complete, the open mapping (Y,X). This follows from the corollary to Theorem 2.13. To my knowledge it is not known if every Hausdorff quotient space of a -complete space is B- complete and so we cannot prove an open map ping theorem for B-complete spaces. Strangely enough, this problem is related to the problem of whether every B — complete space is B-com plete. V.’o shall prove that the following two assertions are equivalent: (i) every B — complete space is B-complete ; (ii) every Hausdorff quotient space of a -complete space is B -complet This result is given as an example by Schaefer 29(c). in [161 p 198 Ex. Before proceeding we remark that we shall use results on duals of subspaces and quotient spaces given in the appendix sections II and Suppose (ii) is true. Let M dual is Then M 0° = M , the W(X' with the topology K : X ->■ Y Y be a B -complete convex space. , X) X' , viz. X'. Let Y * X/M° under is B -complete, and its continuous closure of W(X' , X) its as a subspace of Let X be a nearly closed subspace of the quotient topology. X' Let and Y' M in X'. We endow with the topology it inher W(Y' , Y ) . Thus be the canonical mapping, then M is dense in K* : Y 1 -► X 1 Y is the inclusion map I. X' K K1 - I Y=X/MC Polars taken in We shall show that neighbourhood in (K(U))* = (K 1 ) which is W(Y * , M Y) " 1 W(X* Y M Y Y'=M 0 0 or Y' will be denoted by the symbol is nearly closed in where U (U°) = U° n Y*. is a neighbourhood in Y' ,and so is closed in K(U) be a X. (K(U))* O m - U° H , X)-closed by hypothesis. Thus Since Y is B -complete, X' , and X is B-complete. closed. Let *. m , (K(U))* n m M = Y' = M 00. is Thus The required reverse implication follows from Theorem 3.7 Corollary 1. Let ^ (B) B be the class of B^ -complete spaces. includes the barrelled spaces. We have shown that It remains to prove the reverse inclusion, that a convex Hausdorff space is in (B) -6 only if it is barrelled. It vector space Proposition A is an absolutely convex X and 6 p is the gauge of {x € X:p(x) < 1} C a C {x We shall show that if {x : p(x) < 1} C x G a . We X x € Aa } • 1. convex, x G Aa for all A> n X:p(x) < l}. ieed only show that if Now inf{A > 0 * - n±L 6 then, by 115] p 13, is a topological vector space, . a A absorbent subset of the 1 . Since For each p(x) = 1 A is absolutely n choose n -» «. Hence n an* Then A a G a 0 an = H+T x It follows that, if then x as is closed, x G & A - {x & X : p(x) < l}. (It is interesting to note that this last result holds for the with closure of A in any vector topology for Emfnoo, on x tsa convex space such •1 0 ^ ZkmacA % % % % /bribery Let B X. y, Let « ^ ^ SlnC= rglKIB!) r n Let. x £ ) 11 X/M. X/M. We and Let q , if K,B, M1 X < | and V < 6 «• Then x e 6 b . o , inf{X>0 „ 4 M € Xk,b) ) such that x - Xb + so 13 Thus V . convex. xE B and 1 implies that in m x t M C e , then as a mapping from V and 2 N in X. X V - cl (K(B)I K(B) n x/M into the Banach space since B B show that number. , y - K(x). Choose theclosure x/M z C of is dense X with G in in Y. G. Choose Y r K(t) + e.v and t G x + u mappi G of X is X * Y. We need to Let e be m y positive K(z) G y - K (x) + e v ............... .. Let Ubo an absolutely convex neighbourhood in meets is a union is a neighbourhood in X By hypothesis, it is sufficient to show that the graph closed in X x Y. g and' If we can show that K iscontinuous as a from X .into Y.then we can deduce that Lot (x,y) e B , ). We notice thatB - x " (X(B), . x " (Vi , of cosets of x/M x. (see (15) p 108 remarks following Corollary V. K(B) is q-closed in x/M Let , e B . so « t „ e B - is the closed unit ball in the Banach space K Thus We must therefore prove that be the completion of (X/M , v) We can regard n. is closed in any vector topology on Is a union of cosets of bet and ) , which, by the note above is 1 p(x * n) < p(x> t 0 < by the above note. for We wish to show that v) X/M, show that B ia V 'ClOSed in X ' “ d "= shall show that B by showing that it is a union of cosets of i n X. " in M) = 0 denote respectively the X/M. , and M be the canonical is absolutely (x/M K(B) . < , ♦ „ < = x/M , r(x + M K(B) M - ke, p , r(x + shall r(« t quotient and norm topologies on equal to Suppose x e is the closed unit ball in c& „ . g(B) is « Se.i- K t X -t x/M ? b C 6 b , sinceB is , n o m on X - then , then clearly >0. „ ,b e „ 6 (X,Y) is shsolutely convex and absorbent then we can deduce that There exists ln Let is a semi-norm on x + M e X/K. Hence e B X. K(B> that tarreZZej. be a barrel is a closed sobspace of maPPln9' X.) with X. Then <x + U , y + EV) K(t) G y + £:V , then } (x) C K(t) + K(U) and so y - K(x) G K(U) 4 EV. From (i) , K(z) G K(U) + 2eV , and so K(U) + 2l K(B) = 2el'-.( cosets of that M in X z G 2EB = 2EB. U + B) . and so K(z) € K(U) + 2CV n %/M = Now z G U t B u + 2EB. From this it follows From (i), y - K(x) Since this is true for every positive of K is closed in X % y We have shown that if is a union of k(2EB) + EV C 3EV. 6 E , y = K(x) , the graph and our proof is complete. B is any subclass of the class of B^-complete spaces which includes the Banach spaces, then is the collection of barrelled spaces. 4> (B) The results of Chapter two tell us nothing new in this case, as it is well known that an inductive limit and a product of barrelled spaces is barrelled and that a subspace of finite codimension of a barrelled space is barrelled. However, the results of this chapter and of Chapter two constitute proofs of the above facts. subclass of the class of Banach spaces, Also, if & (B) B is a proper may be larger than the class of barrelled spaces, and the theorems of Chapter two help to characterise it. We can now give the example remarks following Theorem 2.6. We wish to show that if B class of B -complete spaces, then formation of closed subspaces. promised in Chapter two in the -6 (B) is the is not closed under the To do this, it is sufficient to find a barrelled space with a closed subspace which is not barrelled. Every convex Hausdorff space is topologically isomorphic to a sub space of a product of Banach spaces (see [15] p Corollary). H™usdorff. sp Proposition 19 A product of Banach spaces is complete, barrelled and It suffices, then, to find a complete Hausdorff convex e which is not barrelled. topology 8 8 T (8." , I) The sequence space Jt" under the is complete , by [15] (p 104 Proposition 1 Corollary 2), but it is not barrelled, as This example is due to Kothe [11] p 368(5). T , &) # s(i00 , 8 .) . Q, WEBS A CLOSED GRAPH THEOREM FOR CONVEX SPACES WITH Let X be a vector space. Suppose that to each finite sequence (n , n , n , . .. , n ) of natural numbers there corresponds a subset 1 2 3 K A of X and that these indexed subsets are such that n n n 1 2 K U A , A = U I e N n! ni n G n 1 , ... A "i": <= U n in 2 ,,,nK-l n C N & 1 2 co' 1 x tinn An'f A of subsets of X indexed by the finite sequences of natural numbers and which satisfies the above conditions is called a Web on X. If the sets of the web aie balanced (convex, absolutely convex), we say that the web is balanced (convex, absolutely convex). Let be a web on a vector space X with elements A n n 1 2 ... nv K The following definitions of a "strand" of a web and of a "compatible" web are due to Robertson and Robertson (n^ , n , n sequerce ... ) [15] p 155-156. If is any infinite sequence of natural numbers, the (A ,A , A , ... ) is called a strand of the web n n n nn n 1 Each term in a 12 12 3 strand is a subset of its predecessor. Whenever we are dealing with one strand at a time, we often denote the Kth term of the strand h* a symbol such as vector space, S . (X,T) we say that &)' is compatible with neighbourhood U an integer and a pos tive number n If in X , and every strand X is a topological Tif for every (Jy) of <e>- , there exists such that Xs^ C u. Using the above definition of a strand we now give the following definition of a & [4] . web which is equivalent to de Wilde's definition in If the topological vector .'.pace say that ^ is a (* t r to every strand corresponds a sequence if |p^l < y K=1 p x and (X ,1) ( G is convergent. has (S( ) a web Is}" , we of (**" there of strictly positive numbers such that for each K £ N , then the sequence The sequence (X ) is clearly not unique. In particular, if (|i^) such that < 0 < VK is any other sequence of positive numbers for each K G N , then n 1 P„x„ , K K is convergent if |p | < y K K K=1 From this, we sen that, if decreasing sequence and |pK | < (S^) (X^) for each and x K e g for each is a strand of K. we ca: choose a is positive numbers such that if x^, G K , the sequence n I p x is convergent. This leads to our definition of a K= 1 "filament" of a 6 web. If (A^) is such a decreasing sequence of positive numbers, we say that the sequence of is a filament of the web X We have then, that, if (l'y) (F},) ■ (A^S^) of subsets corresponding to the strand is any filament of the web (S^)- , the n S series K-l each p x K K is convergent if x G p K K and |p | < 1 K for K. We now show that if the topological vector space has a web we can construct a balanced web ^ so is T* of and if , let B on ... n_ K = {Ax : x G It is easily verified that the sets web on X. a B For each set A ... n_ K ... nK ^ since and X thatall webs are is a ^ i 2 *"' K IAI < l}. constitute a balanced X has a base of It is clear that if (*)' is a G From now on, we shall assume have, that if . n n l 2 n n If Ifr is compatible, so is 1 balanced neighbourhoods. We now such that, if (t}' is compatible, X is a g, w e b , so is 'V3 n n l 2 , web so is VV' 1 . balanced. topological vector space with a O', then & web &>• and (F^.) ir; any filament of n y x is convergent if x G F for all K G n . K-l K K k Conversely, if X is a topological vector space with a web Af , and to every strand (S^) of W" there corresponds a filament (Fv ) = ( A s ) , with A > 0 each K , such that K K K K n I x is convergent when x^ G , then 4? is a & web. K=1 This idea of a filament of a web enables us to give neater proofs of most of the theorems concerning If ^ is a 6 web for (X.i) 6 -webbed spaces. and (S^) is a strand with corresponding filament balanced and (SK ) (X ) ' since the set (F^) “ is a decreasing sequence of positive numbers, and is a decreasing sequence cf subsets of filament (F^) every neighbourhood (X,T) x U G F K (?%) such that \ U , each has a and every filament Suppose this is not the case. a filament X , it follows that the is a decreasing sequence of subsets cf If the topological vector space K. K. \ Then (t^) X. web, then for 6 ' Fj( c ^ ^or j0mt Choose a neighbourhood FR 4 U K for each x^. K e N. U and Choose is convergent, but x^ 0 , K=1 which is a contradiction. is compatible with From this, we see that a 0 vector space , ... subset X web for (X,D T. Let us recapitulate before pv>ceeding further. (n n arc Sk has a web ^ , nK ) A topological if and only if to each finite sequence of natural numbers, there corresponds a balanced A of X such that X = U n G N n ^n ^ ... n^ A^ i ,... TheSOtS constitute the web . If , n^ , ... ) sequence of natural numbers, the sequence is any infinite (Afi > An n ' An n n ' 1 is called a strand of W strand (S , S , S , ... ) (F , F , (X1) if & . f \ ... ) o f # is a web 6 1 2 r every there corresponds a filament ,where F^ = for each K G N and is a/decreasing sequence of strictly^positive real numbers, and >. G f K K for each K , the sequence \ xR converges. K=1 sequence of balanced subsets of F 1 3 If and only ifto We have shown that w e r y filament of a web ^ in 12 X and (FR) X , and if U is a decreasing is any neighbo rhood any filament, then there exists r < N such that C u. For our next theorem, we need the following easily proved result: if A XA + is a subset of avector space X,y ( C , then 1JA C (|X| •> |M|)h (A) . theorem 4.1. a compatible web # Proof. each and K. Let If X then is a Q, web. (SR ) be Choose x G ie a sequentially complete convex space with F ^ a strand of 6^ and let for each K , and let FR « 7 SR for n n v Lt be an absolutely convex neighbourhood in . X > 0 2 n < X. and r E Then T -T m N such that = x n Xs C u. c . . . . . 2- X . h "* + ... (2 2 _nh(S m)h(Sn+1) + 2 .) C Xh(S n+ 1 Thus (Tr> is a Cauchy sequence, and since complete, (S ) converges and The question as to what 6 when is a is a & is sequentially web for (X,T) vector topologies on is Hausdorff. X web. 6 - webs has beenanswered completely T by the note above C u. n+ 1 and o X , then W is a Q, web weaker vector topology for m > n > r ,+...+ x n+ 1 m ..... 2 " X C It is clear that if Let e c have X i k and choose and U X for is any (X,0 ) . strongerthan by Powel] T [13] in the case We shall prove Powell's result later in this chapter. We new give two examples of convex spaces with & webs. Every Frechet space has < Frechet space X web. 6 as follows. Let We construct {u^} closed absolutely convex neighbourhoods each n. For each finite sequence A e V , n ••• n K n ,U,. j.l 1 Then A n n 1 is of the form (n U , n U 1 1 and so 6 ^ 1 1 is a comoatible U n U 1 n.u. n i I constitute a web . . . n„ K 2 for in N , 2 <x> V j absolutely convex and absorbent, and co co k K-l u a u n n.u. =■- n nK- 1 V , ••• nk nK- 1 j- 1 1 j- 1 Thus the sets with un+^ c (n , n , ... , n ) X = the be a countable base of for X 1 let a web 4X for ' , since U 1 is 1 «*> ( u nvu„)«a nK., ) K K n. " - nK on X. A strand of H n U , n U n n u n n u , ...) 2 ?. 1 1 2 2 3 3 web. Since X is comolete,4? is a C- web. The strong web. Let dual, X* , of {u^: n E neighbourhoods for X. n ) a netrisable convex spc.ee X be a base of closed absolutely For each finite sequence has a <2- convex (n n ... l 2 n ) of K natural numbers, let form a web if is in \ ... X" , s i n L of the form , U» is strongly bounded, " "n ' X" - The u m=l , U ”, ... ) • nK S e t 5 Every strand of W fcr some m G N. X' is complete under this topology, 1 Corollary 1) Each U" (this result is easily deduced from [15] p 71 and so (& is compatible with the topology Lemma 2) Since n is a s(X' , X). (see [15] p 104 Proposition web. 6 The following two lemmas are needed for the proof of the closed graph theorem for spaces with & If LEMMA 4.2. web W each 48 aBair'S topological vector spacewith a X , thee 4J- has strand (SJ n ,n n ......n G N. ^ Now e N , A is not meagre. nj 5 G N , A n n 1 is be has interior for S„ Afi n # X- U n G N N 2 1 n each that n. Proof. Let the sets of 1 webs. A , hence for some = U n G N i not meagre. ' where Afin , i and so for some Proceeding in this way, we obtain the 2 required result. LEMMA 4.3. T Let X and Y be topo) a closed graph linear mapping from for ewei* MeipkbourAood then y * 0. Proof. Let respectively. Let so y = Hence 0 U , W X and er If "• y e W + T(U) neigktowrkood neighbourhoods in 1/ in X , X , Y is a neiy^oourhood of (0,)) wG w , u (u,T(u) )= (u,y-w) G (0,y) G 5 = 6 u. G , where G Then in X x y. is the graph of T , and . linear mapping and if & T ? be balai , where We remark that if under in (U,y + W) y = * + T(u) (U,y+W). W ;.cal vector spaces and X and Y are vector spaces, is a web in of the elements of & T t X ► Y a Y , then the inverse images form a web in X. A further observation needed in the following theorem is tha. i *. X is a topological space and A interior, then there exists a subset of X whose closure an interior point of A with A has a G a. The proof of the following theorem has been adapted from de Wilde [3] p 376 Theoreme. (Closed gxvcph theorem for convex spaces with & THEOREM 4.4. wehc.) web Let and X T be a convex Baire space, a linear mapping from X Y a convex space with a e into Y. Then T is contin uous if (i) or T (ii) has a closed graph, T Proof. has a sequentially closed graph and Let wf be a & web in of the sets of (bY form a web in (S ) of (& such that T- X. (S^) 1 y with Since F n for each y E T~ is balanced, y + T^ 1 T- 1 (F^) n. Let (F^) has an interior ( n• « • • • • (F) is a neighbourhood n in U in X and each n N ,T"' (t^) c~ 1 (F ) + U , and so T(T-> (F I) C f + T ( U ) .................. n n 11 Let V be a closed absolutely convex neighbourhood in Y. 1 Choose m so that (i) that Suppose that the graph of T is closed. We shall show T(y € T“ 1 {F ) . T (x ) e Fm - Choose x Now choose e x ) such that and for each Let T(xq) - T (iym+1) ~ (x^) - I is a (T(x )- T ( 2 - r ,mtr) -T(,r)) 4 I TlF'y^) filament of Cfr and so,using (a)and (d) we see that y ’n Y. We shall use Lemma 4.3 to show that y *» T(x ) . U and W be absolutely convex neighbourhoods in X , Y G T*' m+r ) , and so Let respectively. x G r + T (0) c iw + T(U) ) G 2™rF x* m ^i since (F k) is a filament of the & T(X for all r . (c) t*l the right hand side converges to a point r (d) n r=l (F ) )€’Fm+l - T(«r) e n F N , T(,n)- x 2 such that r ......................................................... Now,for each T(x ) - Tlx^^ From such that T (x ^)-T(kym+2)'',r( G T- > Proceeding in this w a y , choose a sequence ,:r e (b) F C V .......................................... m + T->’(F j C 4V , which will prove T continuous. ni in _ (b) , T ( F M f “ )) C Fn + T(iyn + 1 + T-* (l>n+1)) for each n. x X n. For each neighbourhood T' , then for each n T By Lemma 4.2# there is a strand (S^) (F^) 1 itrisable. The inverseimages under has interior for each be a filament corresponding to point Y X sufficiently large. For each r , (F ) + U. Hence m+r for all r sufficiently large, . web . Also, It follows that T ( x q )-T(xr> - y r T(x ) - y € *W + *W + T(U) C W + T(U). Now, for every C Fm + T( x r ) -* 0 , and so element of T T- 1 (c) (Fj ^m+l^"'+ 2 + 2"’Fm + ‘" + 2 2 by the note preceding Theorem 4.1 c 3V , since and ^ Km +n-l + 2 Fm + " " + 2-1 C 3h(F ) m By "(a) T( x q ) = y. n , by ie), (d) and (a), T(xo) - T(xn ) £ Fm + 2'1Fm+1+...^ Now By Lemma 4.3, V is absolutely convex. T( x q) e 3V = 3V. , T(y^) E v. Since , it follows that x^ T(ym - ‘T ’ is an arbitrary 1 (Fj) c 4V. Thus is continuous. (it) x l 3 Suppose now that metrisable. bourhoods in prove that X From each - T o f for each n. for each we shall as U , n. Using a similar procedure to that of^part <xr> such that " T(xr) E (2 - r ' 1 (i) iJm+r for each r. o and For n G N , Tlxo ) - T(,n ) - j lTlx^,) - T<2'r- 1 lklr„1)-T < » , » + j The right hand side converges to a point (U ) absolutely convex neigh (b) , choosing JUn + 1 + T ( i u |+1> £ Um . the proof, we obtain a sequence Tt,^. (U„) such t h a t V ^ ' - . (Fn) T I U J C 3V. x has a sequentially closed graph and Choose a base T(U ) c T(y ) + Fn " Let T is a base of neighbourhoods for sequentially closed, T,x„, - 0 X. and so y < Y ai'.d Since 1 T< 2 ’r'ly'"«-l'- xn > C , since thegraph T(x,, - y. of Tis The proof that T (x , € 3V proceeds in a similar fashion to that of the corresponding o proof in part (i). COROLLARY. / rtoMVca .'/auodorff spaca with a G web to Mot a Batre apace under any strictly weaker convex Ilausdorff topology. Proof. Let B See note 2.1. be the class of convex spaces with theorem tells us that 6. (B) webs. The above includes the convex Baire spaces and *, (■>, the metrisable convex Baire spaces. Chapter two to this case. 6 Let us apply the ro-ults By Theorem 2.4, we have that < (B> includes the inductive limits of Baire convex spaces and (B) the inductive limits of metrisable Ba re convex spaces, among these the ultra-bornological spaces. Theorem 2.4 has some substance in this case, no for X = ® C. 1 =1 is an example ofan inductive limit of metrisable 1 Baire convex spaces which is not a Baire space, for, if , 1=1 then X x = and U X , and each n n=l X is nowhere dense in n X. (We identify : x. = 0 for i > n } .) i By Theorem 2.6, & (B) is closed under the formation of subspaces n {x G x of finite codimension. I do not know if a subspace of finite codim ension of a convex Baire space is also a Baire space. By Theorem 2.11, (B)( of products of cardinality cardinal if II C.G i G I £ (B) ( -6 , (B)) with I = d. A cardinal (b) if d (B)) is closed under the formation d c = E d. y is said to be where each y cl utvongly inaccessible if < d (a) and there are less than d > N0, d f y 2 < d. It is not known if any strongly inaccessible cardinals exist. The summands in £ d y then c < d , (c) if f < d Y Mackcy-Ulam theorem states that the product of is homological if cardinal. Let I d G C. d homological spaces is smaller than the smallest strongly inaccessible (A proof of this theorem may be found in [11] p 392.) d be smaller than the least strongly inaccessible cardinal and let be an index set with cardinal II i then I = d. By the Mackey-Ulam theorem, is bornological and being complete and Hausdorff is ultra- 1 i bornological (see [15] p 83 Theorem 1, noting that Robertson and Robertson call a bornological space a Mackey space). Thus (B) is closed under the formation of products of cardinality d . We now ask if Tt- is if II C G i G I 1 6 (B) & (B) is closed ubder the formation of products. for every index set I. We give now a proof that every product of complete metric spaces is a Baire space, giving us an affirmative answer to the above question. This theorem is given as an exercise by Bourbaki in [2] p 254 Fx 17(a). THEOREM 4.5. space. Any product of complete metric spaces is a Baire Let X be the topological product n X1 where I. Ls a complete metric space for each i ,f open sets for the topology on is the collection of set. of the form n B.(r> x i e j for each II X X , where J in the index set A base a finite subset of is I and i ^ J 1 i E J , B.(r) is an open ball in wikh radius We call such a set an open hyperball of radius r. an open hyperball is called a closed hyperball. r. The closure of We note that ev, ry open hyperball includes a closed hyperball. Suppose there exists an open subset A of X A and A is nowhere dense for each , n n=l we can choose a closed hyperball of radius A - H and hyperball H n. u C A n A H " A H - *. of radius such that A ^ such that { A, , hence we can choose a closed < ir^ r such that c and 111 Proceeding thus, we obtain for each . ' hcrue A 1 n a closed hyper ball Hn with radius rn H C H . Each H is n+1 n 11 ^ r j such that "n ‘' An " V ^ ,f the form R D (.#n) , where D (i n) is a closed X^ ball in 1 = 1 of radius rn < or D (i>n) - X.. 00 00 »ow " Hn . n=l n (d. = ) V.n,- X. X, . whose radii converge to sero. is ompluta, ' 0„ n=l 1 fhos e„:h “ H ' ' n:l ' n=l 1 j *5 the intersection of a decreasing sequence of closed balls in incc each 2 n 1 is not zero. n- 1 " n , hence It follows that f 4 A. X Let But f f € G „ 1. a Point " . then In this case, The sets c A , which is a contradiction. B Z n W , with linear image then T(X) of convex It iseasily verified that if the convex space web, so does any sequentially closed subspace filament o f „ for is a Baire space. spaces with <= webs. has a G Xv f <t An We consider now stability properties of the class X in » 1 W C # (F^z) form a w e b T in Z. isafilamentofT. of a convex space X with a 3 U Z of X. n'K > Acontlnuous Web V also has 37. a 0-web. (Fy) If The sets T(W) with is a filament of ^ W G ^ , then form a web XYJ’ in is a filament ot 'VJ> . ( T d ’^)) Thus every quotient space of a convex space with a has a 0 web. T(X'. 6 - web also From this, and from Tueorem 2.10 Corollary we may conclude that the open mapping theorem he -■*» for the pair convex Hausdorff spaces if X has a G ' and Y (X,Y) of is an inductive limit or a Hausdorff product of Baire convex spaces. Note <i) If X is the projective (inductive) limit of the finite sequence (X , X , ... , X^) linear mappings T^ : X » X^ (T^ t of convex spaces under the -*■ X » , then X is also the projective (inductive) limit of the countable infinite sequence (X , ... ) , X 1 of convex spaces under the linear mappings 2 : X X^ (Tr : X^ > X) , where = X^ and T^ = Tfi for all r > n. Befor proceeding to consider countable projective and inductive limits of convex spaces with £ Let X webs, we introduce further notation. be a convex space with a web & 6 and let An n ••• n i 2 of (i& corresponding to the finite sequence the element Let n = (n , n , n , ... ) i numbers. 2 strand corresponding to strand,wecall If for each Note (ii) theorem If that in 4 X . .6 . 12 t^ie (F^) is a filament corresponding (F^,) a fi K , we say n n '' * * ^ 12 3 lament corresponding to that A and B (x^) n. is a point sequence of are balanced subsets of a vector Let fJM then X of convex spaces under the mappings converges in Then, if each Ap(A ^ B) ( XA 1' MR. be the projective limit of the finite or Suppose that, whenever (Tr(>'k)) X. If n* 1 O ^ X ^ l countable sequence : X (A ^ (F^). space , and if T be an infinite sequence of natural n. to the above the filament (n ,n^ ... n^) . 3 We call the strand G be k X. for each has a A’r (x i) 6 is a sequence in r , then web, so has (x^) X such converges X. Proof.Without loss of general it \ , v.e may suppose that the sequence for X r sequence (X ) is infinite. , with element A '* ^ n n •.. n. 1 ? k (n , n , ... # n.) 1 2 For each k r , let Y * (' be a Q. web corresponding to the finite of natural .niml-ers. We construct a web (a}" for Then X r.s follows. U B = T‘ 1 n E N nl 1 B n n 12 U B ne N 2 = T' B E N , let B n = T (A ) i n l l Let Then 1 i U (A(1) ) H T " 1 n € N "i" 2 2 U A U) n^ N n 2 (A ) n T ' 1 (X ) = B n^ 2 2 . K > 1 , let = T * 1 (AU ) ) OT-' (A(2) n ) n ...O T - 1 (A(K)) . . . . % n n ...n_ 2 n n ...n„ K n„ 12 K 2 3 K K n n .. .n„ K 2 1 1 In general, if i 2 » T" n in n A (1) = T- 1 (X ) = X. "i 1 1 = T- 1 (A(1) ) O t" 1 (A (2)) . i n n 2 n 12 1 U ne n For each (a Then u b = t" 1 (a(1) n ) o t " 1 (a(2) nK £ N n 1n 2 ,,,nK 1 n i*‘,nK~l 2 n 2 )n...n t"' (Xv ) nK-l « B n r * * nK-i Thus the sets B constitute a web n n ...n„ 1 2 K for X. Fix a sequence 6 ^ n = (n , n , n , ... ) of natural numbers. To show that l 2 3 is a fi, web for X , we must find a suitable filament (If.) of correponding to the above sequence. such a way that if (x^) is a point We shall choose sequence of r E N , (T (x ) : K = r, r+1, r + 2 , ... ) filament of the web ‘V* (s|r\ be the strand of‘V* sj 1 * , ... ) sequence (n^ , n^ r of X . ,...), and let corresponding filament, with (v) K 0 < < 1. Notice that, from corresponding to the sequence SY = T"' n T™1 (s‘/)) n 2 K-l K 1 K Now If*: F Iv F K C T" 1(fJ 1 ^ T * j Is “ K 1 2 K-l (F^2!) n IV — 1 is clear now, that if then for each (x ) IV n X then for each ^ r E N let corresponding to <F|1' , , ... ) the be a = V S ^ , each K , and K K (a) , if (S^) is the strand of (fr ,then ... n t ; 1 (s K)) . K 1 ••• ^ i S K for each K € N. .. . H t" 1 (F K) ) IV ^ Then by note (ii) above. is a point sequence of the filament r , (T (x ) , T (x Jl in is a point sequence of a For each 1 (F^) (F^) X sequence of a filament of the web IT i X ) , T (x IT J f » <<• is]OW ) , ... ) It (F„), is a point V T (x ) i«l r r + 1 is K 39. \ convergent and so is TMx^ for each r. Using the hypothesis for X. i»l of the theorem, we see that A finite or covntable product of convex spaces with corollary. webs has a <2- & Let is a & web (X,T) web , and so does a finite direct own. be the inductive limit of the sequence convex spaces under the mappings : X^ -* X , then (X^) X of is spanned by 00 U T (X ). n n n=l We shall show that of a sequence (Y ) (X,T) is also the inductive limit of convex spaces under mappings sn ; Yn ^ x ' 00 with U X n £ , n=.L X. . s (Y ). n n Let For each Sn : Y^ -> X n , let Y be defined by n be the direct sum sn = T 1 + ^: + ' ** + Tn' i=l 00 Then U X = S (Y ). Let (X,0) be the inductive limit of the n=l Y under the mappings mapping for each n Let and I(r,n) 1 < r < n. : X^ ‘ Y^ Then be the injection “ sn ° 1 ^ ,r^ in each case. I (r ,n Y n S X Each T each Sn (see [IB] p 79 Proposition Each T 0 is continuous wh n has is clearly continuous when is weaker than THEOREM 'jrith C X convex spaces with X Thus T is weaker than has the topology An inductive limit of a eequenc 4.7. Let S). i , hence so is o 0. and so T. webs also las a © Proof. the topology of convex spaces web. X be the inductive limit of the 6 webs under the mappings Tr t sequence + X. (X^) of By the above remarks and by the Corollary to Theorem 4.6, we may assume that to. X = U T (X ). r*=l and let For each r , let &> ( ' be a & web for X , r A Lo the element of ^ corresponding to the j*•• nK sequence For (n , ... , n„) . We construct a web <*?" for X *'• n <= N , let A =T (X ) . If K > 1 , let i n n n l l l (n ) each An n ,. .n 1 2 An n = Tn (An n ...nJ" 1 1 2 K K as follows, i Zt is easy to verify that the sets constitute a web for X. Let S be a strand of (& , K 1 r , S « (T (X ) , T (S ) , T (S ) , ... ) , where r r ri r 2 (S , S , ... ) is a strand of & ' . Let (F , F , ... ) be % k(r) 1 2 a filament of 6 / corresponding to the above strand.As a filament then for some corresponding to S , we choose, (T (X ), r i x T (F ), T (F ),... ). j With this choice of filaments for the strands of & is a w e b , since each web for 1 x 2 , it is clear that is continuous, and is a 6 - . 00 corollary 1 If each . If COROLLARY 2. then (Xji) has a Proof. X h a s a Q web } so does © is a convex space of countable dimensions web. 6 0 be the strongest convex topology for Let X X. Then CO (X,o) = a & © C, i=l web since , which has a t 6 web by Corollary 1. is weaker than 0 (X,T) also has . This last corollary provides us with an example of an incomplete convex space with a of 6 - web. Let p > 1 , and let Z be the subspace consisting of those sequences of complex numbers with only a finite number of non-zero terms. inherited from does have a 6 Then Z , under the topology 9? , is not complete, b u t , being of countable dimension, web. We continue our study of stability properties of the class of convex spaces with Q. webs. Let (X,t) be a convex space. show that there is a weakest ultra-bornological topology, stronger than [13]. t. We shall Tu f for % W» shall then prove an interesting result of Powell If the convex Hausdorff space (X,I) has a Q, web, so does «... is an ct-space, it : >■" _ .1 : rrrr.i To prove this last statement, ~ “ t n r ; •~;j . = . :u T i ° T ij 5 X ij " X art L°n t U l o ^ on X unaer » i = h th « th. maps T. Hence is weaker than the 0 Ct T „ T . are continuous, then w are also continues, spaces let 1 and so X^. 51. lnductive limit of U.D m (r : i e I> be the family ..TiZ:.: set is not empty. The intersection of^t. ^ ^ ** „ ■ same : _ r property. r : r From this and the r » r than _u weakei than z.iB r ; 0. r „.„ b i o l o g i c a l topology : - - r . " . . . ; It is clear that stronger r is r 7 r ; t _ s(X,X ) , l• r X ::: ;3 A:HEEbr." r ::n:= ” “ topology induced by the norm inclusion mapping for each g(A^) a. It is easy to verify that if spaces X; topology foxhave a © X is a (Xt :./ (X, t ) , and if then t 0. a have a G- is any convex (X3a) does not V - { (x^); xK G F , filament of for each positive integer }.Thus xf conve^aes in X , then the sequence ( % K , is the collection of all xf & x ) and , for K~1 |aK | < m <p, •. 1° -> X , all K , then For each ay <p, ( m x>z) G U, ) G . (a) « }, (*K> A U . = 4> B, . (V . (V) ( I K-l a x , where ' * (a}„) = (1, , since the sets in a = (a ) G l°°. * and let a x.. : |a |< 1 is absolutely convex and suppose y G / ^ , we define a linear function be the closed unit ball in for all X is K G k Xsuch that 0 , 0, ... ) We shall show that each Ay Gp . then B Let 1 1 y ■ y . } . Then spanned by U (xK ) c j x, then there exists a filament (F^) positive number and is weaker than (a^.) ^ & are balanced. For T sequences of all filaments of ^I. b .If 4^ each u T«^XA ^ ' y G F Y web. (FR) suppose B be the is the inductive limit of the , then K Let "* X is spanned by strictly stronger than Let (x ) G X 1^ : Let the convex Hausdorff space Proof. point (X,0) Then (J is a e, web for . where Then under the maps THEOHTM 4.8. web and let (x r ) of ) * 6 B ^" and (Ay• .. K a 0 , 1 aKXK C A B (x,)' is bounded and complete. K Let (x ) G y/ K and consider <p' . ; X -*• (&^)*. (xK ; rj (a” ) * S' ','<v X' X Let y G x' , a = (a^) G I'”. OC = ( I a x , y) « K-l K K Then 00 y K=»l a_(Xy , y) K (a, <|)'(> ^ (y)) “ K , since y: X *■ C ) (a> • 7* K. is continuous. 0 ,...) CO ^ \ The series a (x^ , y) is convergent for each (a^) G , hence K=1 ((xK , y)) G V W(2™,) - W(X,X') [15] p 62,Theorem and complete. B( ) <b'(y ) (y) G and so . continuous. Now U , Corollary 2) hence 6 T Thus is <t>(x^ is W(&",&)-compact B (x ^ is (see W(X,X')- compact W(X,X') and so is a stronger polar topology than is T-complete (see [15] p 105 Proposition 3, Corollary ). K Let (X,o) be the inductive limit of the Banach spaces X^ (XK > under the inclusion maps, then is weak-"? than 0 (X,0) is ultra-bornological and . , we By constructing suitable filaments from the strands of shall show that it is a of 6 ^ for K , then y of (2^y^) G V How, — -B 2 ‘" & the topology x. , T web for <X,o). T , let GK a 2 < . For each filament fk* \ yK ) is T Let yK C GK (F%) for each convergent to an element y- where a = (a ) G ^ is defined by (2 KyK ) er U , K < m ? aK - 2n , K > m. A base of neighbourhoods for the topology 0 on X is the collection of absolutely convex hulls of sets of the form U w (XK X B < V ) G XI Thus , where 4 is a positive real number for each k . m / y^, X-l is O-convergent to r ; o-filaments for K X ‘V 6 ^ , then y , and so if we choose the is a C, web for (X,o). An application of the relevant c ’osed graph theorem proves that y = than 1 u. T. Suppose u j■" an ultrabornological topology for The identity map so it is continuous, and so I : (X,u) u is (X,o) X stronger has a closed graph and ronger than 0- A similar argument proves the final statement of the theorem. corollary 1. and tr I f O' in a G U' b f a r the convex Hauedorff cpo'.t (Xti) is a convex Hausdorff topology f o r is a Q, web f c * fX,T '). X with ( T ' may not be comparable with p a r tic u la r , & is a & web f o r n(X,X') , '") and then i'C t t.) 1 v where In X f is the t dual of X. COROLLARY 2. tm If (X,\) is Hausdorff and has a 6 web, then is minimal among the ultra-bomologioal topologies for X. We can now give an example of a complete convex space which docs not have a G web. Let has a O be an infinite dimensional Banach space. Then T (X,x*) is strictly stronger than above theorem (X,T) (X,T) (X,T(X,X*)) web. (X,T(X,X*)) T is complete and (see Appendix I). does not have a G By the web. From this example we can conclude that not every inductive limit of convex G -webbed spaces has a ically isomorphic to © C i-I G web,for (X,T(X,X *)) is topolog with cardinal I - dimension X. (We have shown that there is no Banach space of countably infinite dimen sion, for we have seen that normable, is not a © C. does have a Banach space1 (secAppendix I). ultra-bomolog 1 cal topology for i .) • i=l 6 web and, not being There is no weaker Let X that if and X Y be convex spaces. is barrelled and Y In Chapter three we showed is B^complete, then In Chapter four we showed that if X . . is an inductive limit or a topological product of convex Baire spaces and (X,Y) G A (X,Y) G ^ Y has a £ web, then In this chapter, we show that neither of these two theorems is a generalization of the other. Hausdorff space X , a convex space graph linear mapping from X Y onto Y We shall find a barrelled with a G web and a closed which is not continuous. The example we give is a particular case of an example of Valdivia in [IB). We first need a lemma from another paper of Valdivia [171. Let lemma 5.1. X be a separable convex space andlet be a strictly increasing sequence of subspaces of % X zz U such that X . Jf there is a bounded set A in X m=l m for each m e N , then there is a dense subspace F such that m g <7. F n X, Let Proof. X. is finite dimensional for all (x) , x ? , ... } We choose a subsequence in A each to satisfy n. (F^) G A \ X J\ p i . Let ®j C X m F = x 1 , let F 2 . of (X^) c Xm , and a sequence and *’n £ *n+l ^ such that ^ x^ (yj n , choose 1 Now choose X^ 1 . x_ m_ X^ X , V * X i y A be a countable dense subset of (x^ , x ? , ... To do this, we choose of (Xj so that (x^ , u 2 and choose y E A \ F,. Proceeding in this way, wo obtain the required sequences. Let show that H - <{x P n x 1 + y 1 , x 2 + ±y 2 , x 3 has dimension at most + 1 y 3 3 ,...>> n , for each . n. We shall Let x e H n X Where I x = show that « (xp + ^ yp > ' with “k Z ° ‘ K < n. ^ Suppose K > n. *p(*P + P Then " K % ^ P'1 a contradiction, and so We r.ow show that H n xn C 'n Let an absolutely convex neighbourhood in X. Since . <± v > yn c JU. 1 can choose Hence H If (,n + 1 n Q > n, so that is dense in X. H ^ X , let suppose The set we extend = B U { % .} x - ’x ^ ^ i U . » snJ 1st U be is bounded, the X. Choose n, so that is d” Se ln X ' =° ”e ^ Then ^ F = H. H - X. independent. It converges to the origin in I > n yn‘ 4 s e X X. sequence H yt ' 4 is dense in „ > V"‘ p=l 11 . B D- y,. ••• > to a Hamel base isUnoa r l y of X , Define a linear functional U on X by 1 i e i (V , U) - n , all not bounded on A n eN and so plane in X Let If' (0) n H. F - and (t. , U) - 0 for each i U is not continuous. which is not closed and so Then F « " (0, if' (01 * Iis a is dense in is dense inX , F ^ X and hyperX. F n Xn is finite dimensional. We now prove a less general form of a theorem of Valdivia in (181. we shall need the following result: I , the convex Hausdorff space W(X 1 , XV) we note firstly U let that Thus p. p WCX , on Uauedorff vpaceo, oith x . " t Xi ‘' p. To prove this, , X') is weaker than p. next, foreach is also continuous when is weaker than in an index set under has the topology X ». mapping. Then each Pj by (15) P 38 Proposition has the topology W(X , X >. W(X , X ’). THEOREM 5.2. inugar X P p : X » X. be the projection X i is endowed with the topology X' is the dual of continuous when 13, Xt , then the product topology, w(x , X') , where i e I if, for each X let (X„) p a r a b l e , suck tbat, for ovory positive n , there exists a one-to-one continuous linear mapping y » X n n V.n roof, that Then t h e m io in L isX„ V U ) © M =2 X x M n. y Yl*1! d^nse sub- <2 Yl W space 0 different from subset of I Proof. I which mets every bounded m d closed in a olosed subset oj L. We shall write and 6 ) OO nfl and x, y, z, t shall be tor n=l in the table below. Vector space Topology x; w(v;,vn ) y; W(to' ,nxn) W(L',L) L' Throughout the proof of this theorem, regarded as arbitrary elements of the spaces indicated: x (x1 , X ? y (y1 , y2 » ya ' z (%' , Z* t (t1 ) E & , X3 , ) 6 n vn ) G © X’ n 2 3 t 3 , ... ) e It xv. , t2 U • IlY -*• IIX be defined by U(y) = (X ' l,2(y } ' U 3(y ' Let U and u. ? nx;, " nv; by u.(t) . (u;(t'), uxt ' ) , u x t ' Consider U' , the transpose of U. RY u' (u(y) , z) " Iiun (yn) » z ) - }’(y ) G 6 Y'. n u'(z) = (Uj (z1) , U X Z 2), U M z 3), U' =Uj is weakly continuous and that ©X (y, U'(z)) Thus U' (z ))• n It follows that U x + U(y) . be defined by f(x,y) is a continuous map from linear and continuous and hence f ' Now let Into 1 ' f ! L -> Ilx f is ©X^ under the topoloqtee assigned to these spaces in the tab .. We wish to prove that topology it by W(L ,L') onto W<Il Xn , 6) x^) . show that f'(® X^) , q C (L is open as a map from L f(L) under the topology T under the induced on In order to prove this, it is sufficient to is closed in Let us evaluate p G H f f'(z). L* (see Appendix V ) . Suppose that f '(z) - (p,q) where , then ((x,y),f '(z)) = (x+U(y),z) » (x,z)+(U(y),z) = (x,z)+ (y,C '(z)) = ((x,y),(p,q)) Letting q = U'(z) (x,p) + (y,q) . y = 0 , we see thatp = z. From x = 0 , we obtain , and so f 1(z) = (z,U'(z)). Let M = { (t,U*(t)): t G H X '} and Z = M n (Rx' x & Y ' ) . n n n show that Z = f (©>:'). Let (t , U*(t)) 6 z , then U* (t) G © We and so IV (t ) = 0 U^(Y^) is dense in for each n , by hypothesis,and so (see Appendix IV).Thus finitely may values of Using the fact that f'(© X ') * Z. M «= for all but finitely many values of It {((t1 , t \ I' Hencet G © x' and U* n agree on © remain.' to show tl at and all but (t ,U*(t)) G f '(& x ’) n , we obtain that Z is closed in L'. t 3, ... ), (U1 (t1) , U' (t?) , U'(t3) , ... )) 1 2 Now is one-to-one is zero for n. n t - <tn ) tn IV n. : 3 nx'} c nx* x RY'. n n n Rx^ x I I i s topologically isomorphic to H(X^ - Y'). Now g U' : X' > Y ' is { (tn ,U'(tn )) :tn G x'} n n n n n which is closed in X' * Y' since U 1 is continuous. From this n n n wesee that M is closed in RXV x fly '. Now the product topology For each n , the graph of on RX' xR y ' is the topology W(Rx' xR y , © x x © y ) (see the n n n n n n remarks precceding this theorem). Thus Z is closed in L'^Rx* x €> Y ' n n in the topology V that L'inherits as a subspace of Rx^ x HY^. No V = W(L' unaer W(L* , L) (see Appendix III) , and so Z is closed in L' , L). We can now conclude that f is an open mapping in the sense required. Let RX . H = f(L) under thetopology it inherits as a subspace of Wo shall show that H satisfies the conditions c* ^?mma 5.1. H = f (X x IlV ) = X + U(IlY ) = X + Hu (Y ) . (Again wc regard X i i n * n i n n i as a subspace of IiX .) Now each U (Y ) is dense in X and so n n n n cl H - (cln H ) H h D HX O H = H. (If A and B are subsets i itX ^ i n of a convex space then H Thus is also separable and so it follows that each n ,choose V.(P)= (W ,.W ..... 1 then p E 2 A C \ W (Y^) ,0,0, H H is dense in is separable. , and for each ... ) . Let p H. For , let A = { W (l) , w'2) , W ( 0 P ,since h bounded in P H A + B D A + B.) II . £ = f( ©, X x n=l n X C H. A is bounded in Hx^ , and so A is being continuous. Now CO p 00 n„ n=l Y )= ©. X + n n=l nn=l n n. U (Y ) , and so if n W (p+1) € H p E u .,(Y ) which is a contradiction. Thus W * l>^ <f H . p+1 p+1 p+1 1p By Lemma 5.1 there exists a dense subspace D of H , D # H , such then W that D ^ is finite dimensional for each Let G = f-1 (D). required by in L. by W<IlXn Then We shall show that the theorem. f(U) Clearly , © Xj ). Choose x E u n f~' (D) , and x E u G exists a Ltunded closed subset Let y £ y f G. with B C G n B \ G f' B .Since We may regardL Xr = IlY . l (see H g L <"i b C L. n g f (u) H D. Now suppose there with B ^ G , yEfi b not closed. , and so as the topological direct sum ^©^ X^ Proposition 24). Now > 1 I(G) <' f(B) C D D H h be a neighbourhood f (x) E Then there exists a positive [15] p 9? U is a finite dimensional subspace of which is a contradiction. integer q , such that f(y) ^ D , and 'H H. Hence f(y) E D , The proof of the theorem is now complete. Let us consider a particular cast of the above theorem. let L in the topology induced on it such that of is the subspace of Let is dense in B G G # L. is a neighbourhood in p. We X = Y >=Ji!' for each n , and 1eL U : Y X be defined n n j n n n by U ((y )) = ( V ) i where the sequence (y ) is an element of n m m m m Y . U is clearly continuous, one-to-one and linear and U (Y ) * X . n n n n n Let x = (x ) E X and let E be any positive number. Choose K m n 00 I | so that m=K m < K , y | ' < t 2. Let m = 0 , m ^ K. (ym > E y^ be defined by y = mx , ‘ Then III) (y )-x II ( ) |x |2)' < E. m=K m It follows that Un (Yn) is dense in Xfi for each n. The conditions of Theorem 5.2 are satisfied with X = Y = X,2 . n n II L' = X'x n n-1 Thecontinuous dual © Y' . n=l n of L G* ^ for G* L be 0. L , strictly weaker than and Hausdorff. \ L' L in a closed subset i, (L,G’) rits from L' 0 We shall find a convex topology such that , the dual of G’ is 0. Let Then A is also a so A is a be. G*. 0 the polar of Ac is compact is W(L' W(L* , D , L) barrel and A in L' compact, (see Appendix III). We now have is barrelled and , then (L ,X ) 0 G' O is Let T = T(L,G') be a Hence x is thepolar of Let A in , L) closed since A 6 n G' A , barrel in neighbourhood. W(L' Thus (A0 H c ' )0 “ is a is W(G , L) (L,G) is barrelled. the example we need. Hausdorff. A" under the topology L. A A” H G ' bounded and closed. neighbourhood and so (L,x) is barrelled is a dual pair, since, X is strictly weaker than A° be L' different from then A» n L*. Let the topology on T L ^ of which meets every bounded and closed subset of of n=l the usual We also let X' with and we can choose a dense subspace n=i Xn Y^ with X^ = Y^ = the topological product of i- is If we endow X' and norm topology, then of course by (L.O) The identity map has a & web and (L,x) I « (L,x) -> (L,o) has a closed graph but is not continuous. (L,0) web whicl provides us with an example of a convex space witn a Q, is not %^-complete un< r any stronger convex topology. (L,D is an example of a barrelled Hausdorff space which is not an inductive limit nor a product of convex Baire spaces. A SOME OTHER CLOSED GRAPH THEOREMS - SUMMARY In this chapter, we give a short account wit .out proofs of other closed graph theorems. Many results have been obtained by studying a particular class of convex spaces. 4 (B) h where is We discuss some of these first. In 1956, Robertson and Robertson [14] showed that if B is the class of inductive limits of sequences of B-completc spaces, then (B) includes the convex Baire spaces. A convex Hausdorff space W(X',X) boun led subset of X X' Proposition 1 Corollary 1). is barrelled if and only if every is equicontinuous (see [15] p 66 Using this result various generalisations of the idea of barrelledness have been made. Two of these are used in the following two papers we discuss. Let B be the class of separable B^-complete spaces, the class of separable Banach spaces, the convex space C[0,1] under the topology of uniform convergence, and A {X : X X' that (B2) = & (B ) = 4 the class W(X',x;-bounded metrisable absolutely is a convex space and every convex subset of be is equi-continuous}. 4 < B 3) - A. In 1971 Kalton [8] proved That 6 (Bz) = £ ( B 3) follows easily from the fact that every separable Banach space is norm isomor phic to a closed subspace of of an element of & (B) C[0,1]. Kalton also gives an example which is not barrelled, showing that {b -complete spaces} * & [separable B^complete spaces}. In 1976, Popoola and Twaddle [12] characterised the class where B is the class of Banach spaces of dimension at most define a subset A of the algebraic dual X* to be essentially e ^ a r a b U set. essentially separable X -6 (B) spaces. £ By an example they show that spaces of dimension at most c). (B) , They X W(X* ,X)-separable 6-barvelUd W(X',X)-bounded subset of Their closed graph theorem states that o. of a vector space if it is included in a They call a convex Hausdorff space & X’ if every is equi-continuous. is the class of k. bar relied (Banach spaces) # They also show that ^ (B) (Banach is closed under the formation of completions and subspaces of countable co-dimension. Other techniques have been used to investigate -t • lyahen 17) investigates what he calls the CO or spaces.X is said to be a CG space if (X,X) > * Clearly a barrelled Hausdorff, B,-complete space is a The author shows that (X,T(X,V)) is a (X.WIX.YI) CG i= » == space, and that if X CG - space. 1£ anJ °nly “ is a CG space then it is the topological direct sum of two subspaces if it is their algebraic direct sum. Powell 113) has applied a general closed graph theorem of Komura [10] to many special cases. bet convex spaces, and for each „ which makes each T. i £ I , let T. tc a family be a linear mapping from X , then the final topology on a vector space by the linear mappings {xi : i 1 I, T. X determined is the strongest convex topology on continuous This definition is X the same not as the one we use for an inductive limit topology, as it omits the condition that X is spanned by ^ <’ ^ T.lx,). of convex spaces. if T o a a (V,o) Powell calls a convex space be a property a an space is the final topology determined by a set of mappings i X space Bet - y , Where each (X.T) X. , and for each topology for topology by X a space. For each convex u , he shows that there is a weakest which is stronger than T5 . in the case when is an t , and denotes this In Chapter four, we modified Powell's definitions o is the property "Banach". According to Komura', closed graph theorem, the following statements about a convex Hausdorff space (i) for every <Y,T) a are equivalent: space X , (X,Y) % J (ii) for every than convex Hausdorff topology r & 1 , we have T Tq lQ on (iii) for every 1 If an a (ae) property convex Hausdorff topology a , we have " • convex Hausdorff spac space. Y on satisfies (i) Y weaker weaker Powell calls Powell considers four special cases for the a « a = universal, « - barrelled, a - nonnable, « = Banach. The technique used in this paper is to fix a collection of domain spaces for the closed graph theorem and examine the corresponding collection of range spaces. We mention a few of the interesting results Powell obtains which characterise (a©) spaces for the above four values of (a) (X,T a. is a (universal & ) space among the convex Hausdorff topologies for (b) (X,T) is a (barrelled & W(X' ,X) -dense subspace H Tis minimal T. ) space if and only if for every of is the quasi-completion of if and onlyif X' H. , we have that X' C t f , where (For a definition of the quasi completion of a convex space, see [11] p 297.) This is a generalise tion of Theorem 3.5. ( } (X,') is a (n°rmable & Hausdorff topology ) space if and only iffor every convex for X weaker than T every t -bounded set is T-bounded. (d) is a (Banach Q. ) space if and only if for every convex (X '' topology T^ for X which is weaker than absolutely c.-nvex subset of (e) If (X,T) is a (Banach X is T , every T-bounded. ) space with dimension smaller than the least strongly inaccessible cardinal, and convex Hausdorff topology for Hausdorff topology for X T -compact T is not a minimal X , then there is no minimal convex weaker than T. The strongest convex topology for a vector space. I. Let x be a vector space with algebraic dual a dual pair. T(X,X*) tinuous dual a subspace of X. be the dimension of isomorphic to Let is X. X , i must have its con T (X ,X *) is the s = T(X,X"). Then ® C , where cardinal a e A a Hausdorff topology for X X* , it follows that strongest convex topology for d (X,X*) is the strongest convex topology of this dual pair, and, since any other convex topology for Let X*. X is algebraically A = d. If T is any convex induces the Euclidean topology on each C and so each injection map I : C > X is continuous. Now the Ot n u strongest convex topology on X which makes the injection naps continuous is the inductive limit topology, and so the direct sum topology on © C a G A u is the strongest convex topology on topologically isomorphic to © a G p 92 Proposition 23). C^. (X,s) X , and (X,s) is is complete (see [15] a Clearly T (X,X*) s(X,X*) and so (X,s) is barrelled. The collection of all absolutely convex absorbent subsets of form a base of neighbourhoods for mapping T from X inverse image under y T Every subspace of (X,s) L X , and M P » X -> M is closed in Yis continuous, for the of every absolutely convex neighbourhood in X. tion mapping Itfollows that every linear into a convex space is a neighbourhood in a subspace of s . Thus (X,T<X,X*)) is closed. a = (x , x is h o m o l o g i c a l . To prove this, let an algebraic supplement of L. is continuous and its kernel is L be The projec L. Hence (X,s). Only finite dimensional subsets of let X x , ... } independent1elements of X. be defined by X can be s-boundeu. lor, be an infinite countable set of linearly Extend A Let z G x* and z is defined by linearity otherwise. but z is not Bounded on to a Hamel base B (xfi , z) = n , (x,z) = 0 A , and so A Then is z it for X. x ' B \ A , is continuous, not s-bounded. “ , r is finite dimensional. r - - - - — A note on Quotient Spaces of Convex Spaces. II Let M be a subspace of the vector space polar of M in X*. Let z G M° , then x + M = x + M where x , x 1 2 1 2 we may define a linear mapping for all x Gx and all f M* : M° If x + M G x/M , then and K : X •* X/M (M,z) = 0. M be the If -> <x ,z) =(x ,z). Thus, 1 2 (X/M)* by (x+M,f ( z ) )=(x,z) It is easy to show that to-one and maps z G M° and let G X , then z G M°. onto X (X/M)*. We identify (X/M)* is one- and (x + M,z) = (x,z). be the canonical mapping, and let f M°. Let z f M°. X* K' (X/M)*=MC X/M X/M f (z C z = f(z) ° K , Then If now X and K' is the inclusion mapping. is a topological vector space and topology, then z It is also clear that Now suppose (X,T) . X/M is continuous if and only if From this, it follows that space : M" * X* M Then hrs the quotient f(z) is continuous. (X/M)' = M° ^ X ’ , the polar of M in X' (Kl) 1 (X') - M° 11 X '. is a closed subspace of the convex Haustiu- •: K' maps (X/M)' into x X'. X' K' X/M polars taken in and those tak Let K(F°) q , on = (F O W(X/M, in °)* . and X' will be denoted by the symbol 0 , X/M and (X/M)' F is a fin^u by *. neighbourhood in Let T = W(X,X'). in the quotient topology , subset of F n M° is a finite subset of (X/M)1) weaker than X be a basic P' Lghbourhood X/M , where m M°=(X/M)' X/M. Since X'. Then K(F°)™((K1) M° and so W(X/M, q , the two topologies are the same. K(F°) (X/M)') is a is 4 note SwAepoces of Conoac Wawodorff Spaa,* Tivuv von » r— ' fl lot be a subspace of the convex Hausdorff space (X,T), v«» A z . ^ t-ho be the is the topology induced on A by T. ijer. where T f + A° , where a 6 A.' Z j» z2 G X ' ' Thus, we may d e ^ n e a linear by is one-to-one and maps lentify X'/A" and t A ., . (a,z) . I* = x' -+ X'/A" + A* = z for all f \ x V A - , ' easy to show that z If polar of A in X ' . then ( a , z ^ ( a , Y mapping /v _x (A,T ) A'. If % + Let I , A » X X /A G X '/^ ' be the inclueton '' ' mapping, - is the canonical mapping. X'/A°=A' 1' „e consider now the conditions under which the continuous duals oi X and ft coincide. i, and only if A is Clearly W(X,X', is a dual pair if and only if Now suppose that neighbourhood in taken in A p. n A. Now M(A,A') .. or wi ll I'(F) A W (A «A ' ) b e F if end -mly i£ A X if only if is Let f n A a finite subset of W(A.A') (A,* 1 X . be a basic . * ' denited by the symbol . . Since ' ^ TV# and separates the points of is a finite subset of neighbourhood. V. r.1fai i-ViAf- dense in t-W(X.X'). A , where A* X' = A' A , and so U (F ) I is clearly weaker than 5R. A necessary and sufficient condition for the transpose oj a linear IV mapping to be one-to-one. Let theorem. and (XyZ) a linear mapping. Let (YtW) T* : W -*■ Z T ’ is a one-to-one if and only if be dual pairs and T : X be the transpose of T(X) is T. X Then W(Y, V) dense in X. Proof. Endow Y Suppose w 6 w. W z Y W with the T(X) Then W(Y,W) topology. is dense in Y , and that (x, T'(w)) = (T(x),w) = C in the continuous dual of it follows that Y X (y,w) » 0 separates the points of Suppose T' ({0 }) , since Y ' T1 Then x e x. is one-to-one, where Since is dense in Thus w = 0 since 0 and 0 (X ) - (1 ) (id are the zero 1 elements of 7, and W respectively. (T(X)) 00 = {o }° = Y. Thus It follows that T(X) Y , is one-to-one. (T(X)) ^ = (T') 7 T(vT T(X) y 6 y. and so is one-to-one. T' for all and since for all T'(w) = 0 , where is dense in Y. 59. An open mapping theoi'em. Let theorem. (XjZ) j (Yj W) continuous linear mapping from X is a W(Zj X)-closed subspacc of from (X,W(XtZ)) onto f(X) be dual pairs, into Y a weakly and suppose that Than 2. T T T'(W) is open as a mapping under the topology induced on it by W(Y, W). Proof. Lee M = Ker T , then M c be the polar of pairs: (X,Z) M in Z. , (X/M,M°) M is W(X,Z) and Y,W). We shall assume that each on X/M W(Y,W). is equipped with the topology induced By Appendix II, the weak and quotient topologies coincide. Let X/M and • K , where K X continuous. It is sufficient to prove that S : X/M > Y T X/M S’ T'(W) = (I =- S 1) (W) = S'(W) where Z , and so by hypothesis, closed inZ S'(W) is dense in and S'(W) C required. u € m I is the inclusion map from S'(W) 0 , hence is closed in S'(W) is a net in S(X/M) w w ° and so t ^•> t , and S'(W) Now in M° M°. M° is But - M°. which converges to , (S(t^) ,w) > (S(t),w) > (t,S* (w)) , for each w G m Z. is closed (see Appendix IV), hence Then, for each ,S'(w)) for each M° S(t^) S(t) (= S (X/M) . (t is w y so S : X/M -» T(X) (X/M)' = M° , S Suppose is one-to-one and 1! z X K into is the canonical is open. X K onto T S mapping from open, since with the relevant H on it by T(X) Let We areconcerned with three dual of the six vector spaces involved is equipped weak topology, and that closed. w. Hence S : X/M -» T (X) and (t ,u) *> (t,u) is open as 1. nanach, S. dee IfMeatres. Monograije Mathematyczne Warsaw (1932) . 2. Bourbaki, N. 3. do wilde, M. Generalp o T M ^Part 2. Sar U Thiorem du Craphe Fer,^. Addison » e . U y C.R. <1966) . Acad. Sci. Paris Ser. A-B, 265(1967) A376-A379. de Wilde, M. 4 USseaux dans U s Fspaces Uneaires a Sem-nor-ds H=-m. sec. R=y. S=i. liege* Coll in - 8 M M 5 . VectorTopologies.**» da wilde. M. Topological Vector Spaces. 6. Hath. Ann. 1'■■<,, (19/2) 1.7 128. Locally Horvath, d. lyahen, S.O. ConvexSpa,« r School on Topological Springer-vcrlag 1973. Vector Spaces. 7. IS. <1969), no. 2. the 0" ClosedGrIsrael J. Math. Vol 10(1971) 96-105. S Some For,.,s Kalton, N.J. Cambridge Philos. Soc. 9 . o f the Closed Graph Proc. (1971)70, 101-408. Kelley. J.L. and Namioka. I. U n e a r Topological Spaces. D. van Nostrand (1963!. 10. ». Komura, Y. Sci. Ser. A. Linear Topological Kumamoto J. 5(196?) 148-157. Topological Vector Spaces I. 11. Kothe, G. 12. pop m l a , J.O. and Twaddle. I. Sprlnger-Verlag <1969, . On the closed Graph Theorem. Glasgow Math. J. 17(1976) 89-97. 13. Powell. M.H. on Komura's Closed-Graph Theorem. Hath. SOC. 211(1975) 391-426. Trans. Amcr. », the Cloeed Graph Theorem. Robertson, A.P. and Robertson, Wendy. Proc. Glasgow Math. Assoc. 3 (19->6) 1 12. Robertson, A.P. and Robertson, Wendy. Second Edition. Schaefer, Spacee, Cambridge University Press (1973). TopologicalVector S pringer-verlag(1964) H.H. The Space of Distributions Valdivia, H. Math, Ann. 211(1974) Valdivia, M. »(fi) is not B^conplete. 145-149. On B^-cO'npleteness. Ann. Inst. Fourier (Grenoble) 252 (1975) 235-248. Willard, S. Company. General Topology. 11958). Addison-Wesley Publishing 62. INDEX OF DEFINITIONS The numbers in parentheses refer to texts in the list of re absolutely convex [15] 4 absorbent [15] 5 algebraic direct sum [16] 19 algebraic dual [15] 25 algebraic supplement [15] 96 7 associated Hausdorff topological vector space 19 B-complete 19 B^-complete Baire space [15] 73 ball [11] 23 Banach space [15] 60 balanced [15] 4 barrel [151 65 barrelleu il5] 65 base of neighbourhoods [15] 6 bornological [15] 81 bounded set [15] 44 52 CG space canonical mapping [15] 6 closed graph mapping 52 closed graph space codimension 77 [16] 22 28 compatible web complete [15] 59 completion [15] 101 continuous dual [15] 25 convex set. [15] 10 1 convex space 51 6-barrelled direct sum dual pair [15] 19 3 63. equi-con t inuous [15] essentially separable Euclidean topology [15] filament final topology finite-codimension Frechet space [15] graph of a mapping gauge Hamel base Hausdorff [15] [9] [1‘ 1 Hausdorff summand hyperplane [15] indiscrete summand inductive limit kernel [16] linear functional [15] locally convex topological vector space [15] Mackey topology [15] meagre [15] met.isable [15] nearly closed nearly continuous nearly open neighbourhood neighbourhood of a point [15] [15] net t m norm * 1r’1 normable I ^ nowhere dense [15] [15] 112 point sequence of a filament polar polar topology product of topological vector spaces projection mapping quasi-completion quotient space quotient topology seminorm separable sequentially closed 37 [15] 34 [15] 46 [16] 19 [15] 87 [11] 297 [15] 76 [15] 77 [15] 12 [16] 4 [9] 91 sequentially closed graph mapping sequentially complete 6 [16] simple subspace 12 strand strong dual strong topology with respect to a dual pair strongest convex topology 28 [16] 141 [16] 140 [9] S'; strongly inaccessible cardinal topological direct sum 7 35 [?6] 29 topologically isomorphic 3 topological isomorphism 3 topological supplement [15] 96 topological vector space [15] 9 topology of a dual pair [15] 34 topology of uniform convergence [15] 17 ultra-bornological [15] 160 [1 1 ] 34 weak topology of a dual pair [15] 32 weakly continuous linear mapping [15] 39 vector topology web 28 INDEX OF SYMBOLS 3 <A> A [15] 6 A0 [15] 34 a space 3 U topology 3 41 a space cardinal of the set of natural numbers (i) 6 the complex 28 Q, web clxE ^ [0,1] number field [19] 25 [15] 17 52 CG space 6-barrellcd g(A) h (a) 53 /HX [15] 21 [15] 21 the set of natural numbers N n i e i X. 1 i e i Cj ti-1 X i £ I 5 T-a continuous the continuous dual of a topological vector space the algebraic dual of a topological vector space m I TOPOLOGICAL FIRST COMPLETIONS VECTOR SPACFC PROTECT: LOCALLY CONVEX Janet M. Helmsteat HAUSPORFF TOPOLOGICAL VECTOR INTRODUCTION - V ™ ",r." r ? 1 ” Clearly 1 h °— -r M “: P°l"C.lse fixed. •- :z Phi™. nvery m t rl= space „ 1. r; .Lnrrr Explicitly, if f, ,nd f, are iscraetrles dense subspace, of the d e p l e t e .etrlc spaces " " there eXl6tS an 9(fl(x)| fjlx) 9 for all xCM, lsomet' rlc from S, onto 5, and M, such that These results can bo found In (6) E;r“~-~TL“™TT-vzrrdltlon and scalar multiplication can be extended to "neigtecvrhocd- . simply a and subset of Ti:::: X. Then rr : i ith If , A u be a smll L - 0 subset; vhlch contains Every convergent filter on A A converges^o is a Cauchy filter x is a complste convex space, then the subset AA of 1 w iv a u u s e t I JC 1 complete if „ d only If It 1, closed. If X is a convex nrtl • C space, then a subset OVe deflnltlCn lf converges to a point of A of X is complete according to P-lV 1' every Cauchy sequence in A. , rziiL:::" ». , In a neighbourhood In Is said to be r only lf every Cauchy filter on point Of Let M A e, X ,Y If are convex spaces and homecmorphism from isomorphic to Y onto X onto Y, and that T Y. from X is a completion T : X -*■ Y we say that X is a linear to p o lo g ic a l is a to p o lo g ic a lly is isomorphism ^ Also, the complete convex Hausdorff space of the convex Hausdorff space topologically isomorphic to a dense subspace of X if X . x X is Now X is topologically isomorphic to a subspace of a product of Banach spaces (see [41 product and let X p.88 T t X -* Y onto the subspace is t(X), prop. 19 Cor.). Let Y denote such a be a topological isomorphism from T(X) of Y . Since and so it is a completion of Y is complete, so X. In this project, we shall show that, as for metric spaces, a completion of X is unique up to a topological isomorphism which leaves point- wise fixed, and so we nay refer to the completion of denote it X X X and . The above construction of the completion of the convex Hausdorff space has not proved useful in practice. project we develop another construction of Grothendieck [ 2] X due to .Using this construction, we shall derive many results about completeness of convex spaces. and Y In this When X are convex Hausdorff spaces, some of the results we shall prove are : (i) if x has the weak, Mackey or strona topology with respect to X', then k has theweak, Mackey or strong topology with respect to (ii) X' ? completeness of subsets of X is stable under the formation of stronger polar topologies with respect to * the same subspace of (Hi) if t : X •> Y has aunique (iv) if X X . is a continuous linear mapping, then continuous linear extension T T : X -»Y ; is complete then the closure of every weakly sequentially compact subset of X is weakly can pact. The principal source for the material of this project is [ 4 ] ,Chapter VI. A few ideas have been obtained from [ 5 ] - 3 - NOTATiuN we shall often he concerned in the same proof with several dual pairs. In order to avoid obscurity or repetitive explanations, we have devised the following special notation . 1 =% is a base of neighbour^ ods for a convex topology on a vector space say that the topology cf I£ (X,Y) X X on is is a dual pair, and denotes } w(x, „ Let ther.<6 n A (X.V) be topology on which 6 is a subset » The weakest convex makes each is called the ueak topology on and is denoted and then in '■ t dual pair. a X ■ denotes X and (if1 . D ^ denotes of X is a collection of v i then, taking polars in subsets of ^•riA we shall sometimes X y in Ycontinuous i o t e v m m i by Y, w(X,Y). " “‘.r.::rr.iri'’— 3 . c ,, each Z> in >9 is w(Y.X) c c; , .1 £ if and C, 1 •P"« yhe polars in X bounded , , there exists A , B n e e with 0 * , 0 with then oCe/S- > Yof the elements o f » form neighbourhood base for a con-/ex Hausdorff topology for a polar topology for X with resp ect to X. called ? • Combining previous definitions, we call this polar topology Let V. (X,T) AUB C D , » be a convex Hausdorff space with a bast • of neighbourhoods closed under non-seio s-alar multiplication. Then the collection V of polars in ‘M, satisfies the conditions on X. X • of the elements of C,- r . for This topology is denoted U a po.ar topology and is equal The polar topology Let X C z . Y . ^ 1X (X,Y) and (Y,Z) Let o# be a collection of subsets of Taking polars in notation, be dual pairs, with td° X, indeed, it is Z, according to the above is a collecti on of subsets of the collection of polars in X of the elements of . If °^ X is a neigh bourhood base for a convex topology on topology is denoted rn X X, this . Further special notation and also more standard notation may be found in the candidate^ dissertation. 3. DENSE I«t SUBSPACEF (Y ,T) OF ONVEX HAUSDORFF SPACES. be a convex Hausdorff space and subspace of Y. Tx = Then X ' = Y' (X,T ) a dense x T = w(Y,Y ') , then and if (see Appendix III of the candidate's dissertation). Thus a convex Hausdorff space X and its completion have the same dual, anc if the completion has respect to this dual, so does the weak topology with X . The following lemma and theorem show how the neighbourhoods of a convex Hausdorff space are related to the neighbourhoods of its completion. 3*1 . lemma X a dense subset of Proof, of If On X . Let Y ie a topological space, Yt then a e 0 Then 0 and suppose as Now X U^0 of b Y {TT^X) . b not in such that x". There U H (qOx) is is not empty and so we have a contradiction, If the closure in X Y. is dense in the topological space of any relatively open subset of X Y, t) n has interior Y . If and so U and open and is not an interior point meets every non-empty open subset of corollary in a contains a point exists an open neighbourhood empty. 0 c interior 0 A A A is an absolutely convex subset of has interior, then zero is an 3*2 Y, and Let neighbourhoods for Proof 3*1 x 1, and he a dense subspace of the convex Hausdorff a base of absolutely Then the aosures in Lemma interior point c: X is a neighbourhood. theorem space the convex space Y convex neighbourhoods for of the elements of v. form a base of Y. Let UG then U is a neighbourhood in Corollary and the above remarks. neighbourhood in Y. Then vn% so there exists with UCvHx. U€ % Let V isa neighbourhood Then U C v = Y by be a -losed in X, and V. Since Y has a base cf closed neighbourhoods, the theorem has been proved. COROLLARY Proof If Let B X is barrelled, be a barrel in and so it is a neighbourhood in a neighbourhood in Y. But X. so is Y . Y. Then B Hx is a barrel in X, Taking closure in Y, B n x - b . B^lf is ... THE COMPLETION OF A WEAK CONVEX We consider the completion of dual pair. in Now Y* . = (X, w(X,Y) were is a dual pair and X Y or Y* , {o}° - Y* . p.61 Prop. 13) (X;w(X,Y)). X00 = w(Y*,Y) Now and so (Y*,W(Y*,Y) (Y*,w(y*,Y)) (X,Y) is lo show this, consider the dual pair polars in X0° (Y,X) SPACE. w(Y*,Y) dense (Y*,Y) closure of is complete is a . X. Taking But (see [ 4] is the completion of If we assume that the completion of a convex Hausdorff space is unique up to topological isomorphism, it follows that (X,w(X,Y)) For example, if (where — + complete p > 1, = 1) (see [1] is only complete if 5^ X = Y* . under the topology is not complete although it is sequentially p .69 Cor. 29, noting that Dunford and Schwarz call a weakly sequentially complete convex space Now let (X,Y) be a dual pair and endow topology with respect to Y. X "weakly complete"). with a polar In the following paragraph we shall show that the completion of X under a polar topology with respect to is a subspace of Y. Y* - 7 A CHARACTERISATION OF HAUSDORFF SPACE UNDER Let theorem 5*1. THE COMPLETION OF A POLAR Tui- I/. CV . (X,Y) a c o l l e c t i o n o f s u b se ts o f a polar topology on are taken in Then X. be A CONVEX a dual p a ir . L et & be s a t i s f y i n g th e co n d itio n s fo r Y L et ^ +t>° ) 1 where polar8 Y* . X i s a subspace o f Y*, (X, Y) i s a dual p a ir , «9" s a t i s f i e s the co n d itio n s fo r a po la r topology on wider th e topology i f n X i s the com pletion o f X, and X X under th e topology e90n X . Proof. In this theorem, polars are takenin It is easy to show that (X,Y) D E # . { z }° xGx zEX . with zE % + D° . X, so does {z }° 3> w(Y,X) 1 Thus (Y,X) if are elements of A sub-basic where . D° . Thus w(Y,X) Since + D ° Since Y Thus topology, -^° n ^ We now show for some » X on X , since, zE X l x EX X . X , is is dense in and w(Y,X) where and so (£>° n •& , tnen A in and X . Now d (Y,*&• X ) . ' X induces the for each and Let is there exists it is also absorbed by a finite (Xs ify° X) that z X is X . on basic neighbourhood of of Y bounded. D is absorbed by a sub-basic Firstly, it is clear the the topoloqy d *= z - x E x w(Y,X) Jfr satisfies the conditions for a polar bounded. z = x + d (Y,X) absorbs the points of intersection of such neighbourhoods and so Let Y* is a dual pair and so D ° O x zE XD° neighbourhood in D° C\ X . Now neighbourhood in Now z € x j f ° D y. p . topology > ° n Y*. or . We show that the Let is a subspace of X C x C V is a dual pair and and so is X Y D x € z + D° H X . z + D° ^ X is a z £ X. 4 i . O Then Hence the closure = - 8 - It remains to show that a H x Cauchy filter on X is complete. X . The ^ is stronger than the w (X,Y) v,(Y* ,y) Y*. topology on Cauchy fi lter $ on p.61 Prop. 13) w(Y*,Y) topology. and that Now Y* DG topology on X A induced on X by the is w(Y*,Y', complete y* € Y* We shall show that contains an element will tim follow that X converges to a point Let j/*G X , be £ is the base of a v(Y*,Y) Thus Y* . and so topology Let B and in the y*G X + D B C y* + D n A . such that S* ■+ y* (see I 4] X is complete . Q contains an element Let b G b, then b + y* G b ■+ ■h D closed and so Also, B C *3 small of order DU G ^ .Nowb + and so b G x+ >3 D ° A Thus y* G x . From , y4 and b Hence BCy* + D °^X , COROLLARY Proof are in 1. th e topology Proof. (i) , X , y* G b + so is a, and a G D o ^ DnX . Each & is w'l't X) bounded. X has the toplogy sfX^Y) then X has s( Xt Y) . Suppose X has the topology s (X,Y) . w(Y,X) bounded subsets of Then is Y ,each of which A w(Y,X) w(Y,X) bounded. bounded. corollary 3 , Also, Thus every w(Y,X) bounded subset of Y X = s(X,Y). Let X be a convex Hausdorff space v i th a base o f neighbourhoods c lo s e d under non-zero sca la r m u lti p l i c a t i o n . Taking polars f i r s t l y X is X = Proof. by Q y* = b + a, where A % +^D 0 as required. 2 . If the collection of all is w(Y*,Y) This was proved in the course of the theorem. corollary is is y* G x + h D ° hence /» Since A D 1X . S ................................ b G x X + r° . B It X1 and n U G % in (X + % X 1 and then in X ' *, ) under the topology This result follows from Theorem & by the com pletion o f f y ' *^ X . 5*1 with Y replaced - 9 Suppose that the convex Hausdorff topology vector space subspace pair X Y of (X,Y). X*, but that Y* is not a topology cf the dual is a subspace of Y*. to a subspace of X'* under a polar topology with Y X of the convex Haundorff space where ^ Y X*. When is X the whole of Y ? case when A* n y is Now suppose w(X,Y). topology with respect to the w(X,Y) w(X,Y) . p.64 (2). topology on Y. (X,^^ Y) is a polar topology with respect polar topology with respect to [4] (X,T)• X1 . The completion (see with respect to a under a polar topology with respect to and the other a subcpace of respect to T on the Then we have two forms for the completion of One is a subspace of Y & °, is a polar topology, T Y* under a We have seen that this is the X = Y* under a polar All such polar topologies coincide Since the w(Y*,Y> topology on Y* X, it follows that the topology on induces X is T r e l a t i v e l y continuous is Thus T on a is relatively continuous on A. Before proving the next theorem, «« remark that a neighbourhood In a convex Hausdorff space polar of in u y*. t h e n U<= - « ° n v . . Since a linear functional on V V If J and lf the polar of 15 '' U In V , Is continuous If and only it is bounded on a neighbourhood. oology.Let ihie a « * . Men eel r e l a t i v e l y continuous on each Pr„f. • in thi, theorem, polar, will be taken in Y or Y* . r:rz-zz,czzy. Then y 1 » X = Uu ................. if Bet z e V. there exists If in we first show that UE-H ,ei - Y, for each _ to. with zex Z « L ' ° - O' • then for each u0 e Y ' , If and only If for each • O e ^ there exists by the remark preceding this theorm. 0 6.2 , and so z e X . Pe . show n e x t t h a t , i f We u n £>)° c uc + NOW [U n D )° D D° i (u and so NOW y n U° + D° r c o m p a c t and so i t w(Y*,l) , For each z G Y* D G^ zG x yy ,11) . „dy G u n p ««*,Y) * U° 1» M by <1 U since tl>c compact, w(Y'.Y) on Y' P-" (U . 7 absolutely convex. >• Hence • there exists a neighbourhood D G^ Hence z E ( D r , y : ° c u ° u G * Z i such that +D° dii) by . (i i) • D< ft. 2 P-b 1 T h - 6) ' Is relatively continuous on each Conversely, suppose then (Ml l a a :so U° t P° and D) closed,so is C° yez, n u » |(y,z)|< I s and so 2(U ^ and we have proved result Suppose {U^D)° ? U° Y« Induces the t o p o l o g y on U° U 5° C u° * if, and £))0 then ...................................... and u n £> C D , hence f 13 w(Y*.Y) ts DEjf the B a n a c h - A l a o g l u theorem w(V,V) topology c UHDCU By ts D° u e W a n d z^X and so there exists Thus ze >i e (o° + P°) - I (y.z) | < 1 * . and I(P.O l < Let U E 91 with c e (un = • 0 ° z r ( | p ) < (|uJ by (111). zt f o U o “S ‘ e is continuous at 0 In the relative topology on relatively continuous on P and so z Is D . He do not lose generality by supposing that the elements of D are absolutely convex and closed. us a characterisation of suitable collection t 5 when X of subsets of The theorem does give has a polar topolcgy on any Y. If collection of closures of the absolutely convex hull, of^the elements of & are the s a m e . , then the polar topologies A and 6 x COROLLARY it, 1 . r e l a t i v e l y continuous on each on 7, then 2 . r e l a t i v e l y continuous on each Proof dual pair f o r a l l to po lo Y for every topology of the dual (Y,X). 3 . the to p o lo g ies Proof. Jfr r. an a t The to p o lo g ies induced on each w (Y,X) and and a E D .Let z, , in Y. Since z, zn T) , there exists a w(Y,X) y e (a + U) Y have the topology zK, , zn ..... is a basic {zi, DE c o in c id e . In Theorem 6*2, let Let on Y This follows because the same absolutely convex COROLLARY a D e ft on (Y,X) . sets are bounded and closed in of i s a lso contxnuous The same lin ea r fu n c tio n a ls gies o f the dual p a ir by Y which X i s complete under the topology & ° n X . COROLLARY are I f every lin e a r fu n c tio n a l on x w(Y,X). • w(Y'$)neighbourhood are relatively v(Y,X) neighbourhood D ** I(yJz i) - (a,zjL)| < 1 » U 1 < i n | (y - a , z^) | < 1 , continuous such that , 1 < i <n , ■* j/- a E {zi, z 2 , ... » znl Hence (a + U) ^ D C topology induced on it by w(Y,X). w(Y,X) I by a + {zi, z 2, ... , zn]° v(Y,X> However, the w(Y,X) , and so the is weaker t topology induced on it . erthan the topology, and so the two topologies coincide on COROLLARY topology t (X,Y) 4. I f the convex Kausdorff space has then i t s com pletion, X, has the Yackey topi gy Proof, in the theorem, let £ absolutely convex each D in * by Theorem T(X,Y) , w(Y,X)-compact subsets of is also w(Y,X) compact. 5'1, the topology on since every w(Y,X) be the collection of all w(Y,X) % is By Corollary 1, Tak.ng polars in J ° 1* compact subset cf X ^ V are convex spaces and which maps tonnded subsets of X Y is al^o torr*Aogioal mapping from X under certain polar topologies. 6*2 THEOREM tz*, V, then T into any other In the followingsection we shall prove some results about completeness Theorem » U n e . r mapping A convex space Is said to be if every bounded linear convex space is continuous. T = X - * Into bounded subsets of is called a bounded linear mapping. of duals ofbornological spaces For this reason, we restate for duals of convex spaces. 6*3 . Let X be a convex Hauedorff space and U t poZar topcZcg* m subsets of * acZZectio* # ^ ccmpZetion, c k W M, of topoZog* is the set of aZZ Zimsar fbmcticma:* on relatively continuous on each COROLLARY . D in & D i" wf X %' under J:ts wkick era . If every linear functional on reZativsZ* continuous on sack then V*, compact. I£ %, Y. " X whicn vs a&so continuous on X ' is complete under the polai' topology & %, - 14 7. THE CONTINUOUS THEOREM and DUAL OF A Let 7*1. HAUSDORFF X BORNOLOGIC AL be a Hausdorff bornological apace efr a collection of closed absolutely convex subsets of satisfying the conditions for a polar topology on every compact subset of X is included in some X ' is complete under the topology Proof. Let maps bounded sets in follow that z Jfr X D& & . Suppose that z since, if U for each {o, A = n. Now Then z We shall show that (x^) ~ x, such that A CD into bounded subsets of C . of 0 x^-NU 1 . If ••• ) • for some X as complete under X Then £>€$’ . C . It will zc X ' . X. with n -► 00 , X, for all n. A is compact, Now z D n. X is a Hausdorff homological space s(X',X). is a metrisable convex space then sfX'jX) is This contradicts the fact that X ' is complete under the topology particular if z D and so maps bounded subsets of for all corollary is u relatively continuous on | (i- x^, z) | > n N xi , 2^* 2^' and, by hypothesis, X' . is any absolutely convex neighbourhood in n>N**--x E n n Let then is not a bounded functional on there exists an integer Hence If De w and hence that X Then there exists a bounded sequence |(x^,z) | > n* X' into bounded sets in is continuous on X '. X . z € m , the completion of relatively continuous on each then SPACE. and if X In X' is is a normed linear spacet ie a Banach space under its strong topology . Proof. Let in the theorem be the collection of all absolutely convex, closed, bounded subsets of every compact subset of X is bounded, the theorem are satisfied, and so X1 is compl / Since * cions of the a under 6 (X1,X) COROLLARY 2. If X is a Hauedovff homological space, X ' is complete under the topology Proof. Let absolutely cor vex is the topolocy w(X,X') theorem, t ( X 1,x). in the theorem be the collection of all w(X,X') T(X',X). compact subsets of X. Every compact subset of Then X compact and so is its absolutely convex h u l l . X' is complete. & v is also By the - 15 COROLLARY 2. If X ie a Hauedorff homological epaco, X ' ie complete under the topology xoof. aben Let & cly convex is the . pulog>r w(X,X ') theorem, X' t(x',x). in the theorem be the collection cf all w(X,\1) t(X ',X) . compact subsets of X. <5" ° X is also ipact and so is its absolutely cor vex h ull. By the is complete. Every compact subset of Then — 16 — 8. A THIRD CHARACTERISATION HAUSDORFF OF THE COMPLETION OF A CONVEX SPACE In this section we obtain a characterisation of the com pletion X of a convex Hausdorff space show that couple .eness of subsets of X ation of stronger polar topologies on subspace of X which will enable us to is stable under the form X with respect to the same X* . theorem collection of 8*1, u(YtX) Let <X,Y) be a dual pail' and lei & cloeed -ibsolutely convex subsets of satisfying the conditions for a polar topology on the completion of /S % X under this topology. if and only if z (z (0 )) ft D is X z£Y* . w(Y,X) Endow Y with the topology is relatively continuous on each and so (z_1(0)) n p be Then closed for each Let D in Since H = z 1 (0) Z> € «&■ . Let z G x, by Theorem 6*2, D is closed, this if intersects each z vanishes on D , so suppose is relatively continuous on on D . w(Y,X) . X. Conversely, suppose in a closed set. D . is closed in set is also closed in a€ D Let Let . Proof. z X. Y —1 D € «fr then be a z D D in •5’ , then does not vanish Let e be any positive number, then we can choose a point with (a,z) = a , (b,z) = &/=0. If |p| < £ , choose D . It jd| > 6 , also in For, let b € D 0 < a < e . let a to be |6| b , be such that which is T a = £ b G D . 8 Now hypothesis. U in Y a ^ H not meet ing |{y,z) | < a < e this is not true. exist y iG u f1 D + a G 2D and Since 0, and so which is closed by H f> 2 C . for all yi+ a G h , and so z G x y G U'^D . Suppose Uf>D is absolutely convex, with (%i,z) “ - a . which is a contradiction. at a ^ H f* 2 D , Hence there exists an absolutely convex neighbourhood with a+U Now and so Hence by Theorem z Then (yi+a,z) there must = 0 , y\+ a G H f ' 2 D n ( a + U ) , is relatively continuous on 6*2 . - 17 COROLLARY completion Let 1 - X. Let z**", v'rojnu0 V in =ei be a if and on!.* if U(X',X) is closed in X . Proof. In the theorem, we let collection of polars in in then X X' Y be %' and ^ be the of the class of all neighbourhoods X. lf y z"‘ (0) in is a vector space, and if is a hyperplane in is the kernel of Y A subset A w(X',X) X z ^ 0 ,then Y.Conversely, every is said to be closed and hyperplane a non-zex- linear functional on of the continuous dual Hausdorff space is z^y* X' Y. of a convex ^ nearl* ctaeed for every neighbourhood if U A ^ in X. We givv now a useful characterisation of complete convex Hausdorff spaces. 2 . The convex Hausdorff apace COROLLARY if and X if ever* nazrZ* closed Ayperpiame in closed. Proof. Endow X' is complete. Let where , z€x'* With the topology z*1 (0) is complete^ %' w(X',X) . z (0) closed. is closed and Let X is closed. zEx z : X'+ C . Then by Corollary is continuous and so % X1 is 1,z ’ (0) ^ % - X' (The above characterisation of complete convex Hausdorff spaces leads to the definitions of B-complete and 8^-complete spaces. The convex Hausdorff space (i) is w(X',X) (ii) subspace of X is said to be B-complete if every nearly closed subspace oi X closed : B r-complete if every nearly closed, w(X \ X ) dense X1 is X X* Thus Now suppose every nearly closed hyperplane in w(X',X) Suppose be a nearly closed hyperplane in By Corollary 1, z ^ x - X - X" . is continuous and i* w(X\X) closed.) . -18 - Our next theorem compares the completions of a vector X under two comparable polar topologies. If subset of jfis a filter on a vector space X If then ^ ^ V, then is complete in Y. A Cauchy filter on complete in A Y A Y. Let A. Y. Then FE X if and only if X. ^ n X -- a*? A, then Conversely, suppose A. If "* a G A, then "* a » an^ 8*2 . Let (X, Y) satisfying the conditions for a polar topology on Taking polars in of absolutely convex w(Y,X) closed subsets let M, N 1% the topologies £ ° X D If z€ n, then , by Theorem COROLLARY 1. A ° ^-complete in Proof z"1 (0) n£» then is -& x . w(Y,X) closed 8*1 and the fact that#5 C £ If also, K X, it is also complete and hence closed in M, and so is a subset of N. X which % °KX-complete. K Thus K is closed in M This topology is weaker than the topology is closed and hence complete in . z f M . under the topology induced on it by the & 4 °lA X . X under ^t , By the note preceding this theorem, under the topology X. is complete under the topology By a second application of Theorem 8*1, is If ^ is also complete under the topology Proof. for each be the completions of respectively. In particular, if , it A be a dual pair and let A of ° n x is is a base for a Cauchy filler be collections ^ A X and t N C M . is a X. THEOREM Y } . Let Then / f'X be a Cauchy filter on which contains is complete in A. Let : of the convex is complete in which contains f is a X which contains A. is complete in which contains on X is complete 'n Y and A defined to be{a H Suppose be a Cauchy filter on > a, and is a is a subset of the subspace Hausdorff space A H X N topology on M. 'N , and so K is K is complete in X - 19 - 20 - 9. h GENERALISATION OF - THE BANACH-ALAOGLU THEOREM. The Banch-Alaoglu theorem states that if neighbourhood in a convex Hausdorff space w(x ,X ) compact. topology on X' neighbourhood Let x, then is a U° is r'e shall find the strongest convex convej under whjch U U in (X,C) u° is compact for every X. be a convex Hausdorff space and let bo the the collection of all closed absolutely convex pre-compact subsets of X. Then the topology,# theorem with topology C Let 911, 11(C) compact. on X X X' is denoted 11( C ) 11(C) . be a convex Hausdorff space For every neighbourhood . is , on U in X, U° is the strongest polar topology under which the sets U0 arc compact or even pre compact. I'l'oof „ Let be the collection of absolutely convex, closed, pre-compact subsets of of polars in X' then w(X',X) U is By Theorem cf neighbourhoods in U° Now •« c = C 1^1* i s c/t° is u° X. if since it is is also H(C) u° E 73 w(x',X) compact. complete. 11(0 pre-compact. and .'ince each ' « 11(C) . A 1 is 13 We use T h . 3 Cor. p.51 pre-compact, each u° G 13 pre-compact. Let 4 ° each complete, 8-2 Corollary 1, We now show that X, and -/3 be the collection ii e ft is be any other polar topology on 4o ° pre-compact. pre-compact, and so -6 ° Then each is weaker than crt ° . I X' such that ce 4 is ^ ° - 21 10. AN EXTENSION OF A CONTINUOUS LINEAR MAPPING. In this section we show that a continuous linear mapping from one convex Hausdorff space into another can be uniquely ext-ended to a continuous linear mapping from the completion of the first space into the completion of the second. Using this result, we shall show that the completion of a convex Hausdorff space is unique up to topological isomorphism. lemma 10-1 . Let dual pairs and T\ maps nap Z (XtY), T : X -*■ U V into Y. into W. (Y,Z), (U>V)t (VtW) a linear mapping whose transpose, Let the transpose, Then, be if where pjlars are taken in T " , of T' yl C x, T " ( A 00) C (T(A))00 , Y, Z, V * W. 0 Proof. T X Y Z | | T' | T" U V W Since T'(V) e y , T iS (see [ 4 1 p.38 Prop.12), and so (see [4] p. 39 Lemma T'(T(A))°)0 D AC° . 6). W (X,Y) - w(J,V) continuous (T(A))° -(T1) 1 ThusT 1(T(A))° C Now T " ( W ) C Z w(V,W) - w(Y,Z) continuous. Hence 10*2 . Let spac&sand let Then T oo (T(A)) It now follows that X and Y T has a unique extension Proof. X Let U neighbourhoods in be the transpose of into T" is X' . be convex Hausdorff be a continuous linear mapping of linear mapping of Y' so (T(A) )00 . theorem Y. A° and (T'(T (Al)°) ° * ( T " f using the same reference as above. T" (A00) C and (A°) X T X into whi<?h is a continuous into Y. , 'Y* be bases of closed absolutely convex and T' . Y respectively. Since T Let T'* : X '* > Y '* is continuous, T* maps - 22 X'* | T'* Y' Y'* All polars will be taken in For each U G by Lemma X', X'*, Y' or y«* 10*1 , 00 T'*(U°0) C (T(U)) (a) .00 NOV. C [12 ^ I '*< x t 0 T '*<X > I C u e V (Y * (T(U)) UG'ZL (Y + ( r(u) 1 )• such that T(U) CV, (by Iheor.™ ) by (a) bet hence 5-1 31 and the fact that T'* jx = T, V € 't* . z 6 y + v00 , Corollary There exists ufe'lt and so + Let V " O Let UG by(a) be a basic neighbourhood in ^6 be such that and linear map X y .. (b) . from X is dense in X, T(U) C v . T Y T and open as a mapping onto also, T in the case when The X or X onto X onto 10*2, if T(X)t then T 7(\). All closures T(X) . sets T(U) , U G & T(X) . since is one-to-one We use the notation of Theorem 10.2. T(X) ^ Also, dense in T With the hypothesis of Theorem 10*3. will be taken in T. f ($) is a topologi-cal isomorphism of Proof. is a continuous and it is an extension cf is a topologio'd isomorphism of for T - T'*|_ is unique. We now consider theorem VG T Then T'*(U00 n x ' C v00 O y It follows that into Y, where form a base of neighbourhoods T O O " - T(X) D 7(50 = T(X) , and so By Theorem base of neighbourhoods for 4-2, the sets T(X) T(X) is T(U) , U G £6, form a Let u e u and x € x show that W£ % with dense in WC u such that x+W 1 (X) = 2T(U) , is continuous. so . We shall case,wemay X Z/e x + W . is Then Now T(jy) = being closed in T(X). choose X and T This is a contradiction x G 2U . It now follows that and so v,ith 2 T(U)". U a topological isomorphismonto and so (U) does not meet 2U. y€x T(y)ET(x) 4 T(U) C 2TTuTn t For, it this is not the £, S" we may choose y$2\l, and T(y) € € 2U . x be such that T(x) € x Suppose T is one-to-one and its inverse T(x) = 0, then » O. Also, for each T-1 : T(X)X x r"2U U E % ,T for every (fTuT) C UG^ 2U, and is continuous. Let X be a convex Hausdorff epvce and let be the completion of X as defined in Theorem COROLLARY. Let Y be any other completion of logical isomorphism from fixed. Proof Let T the dense subspace Y. Then By Theorem T(X) . T, as X Then there is a topo X pointwise Le a topological isomorphism from T(X) T 10*2 onto maps X into is a topological isomorphism from both closed and dense in T, X of the complete convex Hausdorff space is complete, oo is is an extension of Corollary 5*2 Y which leaves defined in Theorem 10*3, Since X onto X. X Y , and T so T(X) . Thus . 'X) * Y. Y. x T(X) Since 1 .aves X poirV wise fixed. This corollary justifiesour use of the phrase com pletion o f the convex Hausdorff space x" . "the onto is r 3. - 24 COMPLETIONS OF METRISABLE A ND NORM ED CONVEX SPACER. in this section, we show that the completion of a convex metric space is a Frechet space and the completion of a normed linear space is a Banach space. We show firstly that a continuous seminorm on a convex Hausdorff space can be extended to a continuous seminorm on its completion. x f p space hood "p X, then U in p is the gauge of an absolutely convex neighbour X. Taking polars firstly in be the gauge in closure of of is a continuous seminorm on the convex Hausdorff U in X of U°° . Now U°° X', then in^ X, is the oo X* , U let w(X.X') is the closure Since the dual of X is ,oo is a neighbourhood in X. By Theorem 3.2, U U. X. Let z € X, then iniCX > o p(z) : z e x u00 } inf{X > O : | (z,y) | < X inf for all } n {x > 0 : I(Z,y) | < X} yeu° inf ( |(e,y) |, ”) ^ Z/Gu° (i> sup { I [z,y) | : Z/e U°} By the same argument, i inf{X > 0 : x€XlU00nx)} = sup { | U,y) \ s y^V°} p(x) Thus is an e p .oo ision of — U unique, since is dense in If X X is a norm on (1) , subset of z * 0. B p A is a neighbourhood in x X. X'. for all Hence The sets X Suppose Xti, y^b° (z,%) = 0 p is p, let Wt show that z^ X . inen which is an ab«orbent for all y G X ' and so X€ c, form a neighbourhood base for the and so the sets XB X, and A and X. p(z) * 0, where , X€ C , neighbourhood base for the topology on the topology on is X. is the closed unit ball in (z,y) * 0 topolog'/ on Also, is a normed linear space with norm p r g(B) where by p. ii I continuous since p xE X , X . form a Thus is a Banach space. p determines - 25 If X is a metzisable convex space, then the closures in x of a countable neighbourhood base for * a base for X, and so X is a Fiechet space. X form a neighbourhood We have shown that if the convex Hausdorff space A has the weak, Mackey or strong topologies with respect to a subspace Y of X*, so does normable or barrelled, so is X. X if X is metrisable, (see Theorem According to McKennon and Robertson 3*° [3] , it i- Corollary). — known if the properties of being reflexive, semi-reflexive, bornological or u l t r a -bornological are spaces. inherited by completions of convex Hausdorff 12. E B E R L E I N 1S THEOREM. We shall now prove a result about weakly compact sets in complete convex Hausdorff spaces. We need the following lemma. lemma that If A 12*1. is a subset of a convex space such every sequence in A has a cluster point, then A is pre compact. Proof. Suppose A is not pre-compact. absolutely con\ax neighbourhood sequence (xi ,x2 , ... , x ) , n U A such that for every finite n U (x + U) . Let xi € A . i=l 1 Choose xi G A such that x 2- xi ^ U .Then Choose xa G such that xa - x i $ a U Proceeding thus we obtain a sequence that xn “ xm f u for all There exists an n y m. A f (xi + U) and (x^) xa - x 2 f U u) U . of points of A such This sequence has no cluster point which is a contradiction. corollary. then A If A is a subset of a complete convex space is compact if every sequence in Proof. A has a cluster point. This result follows from the above lemma, the fact that the closure of a pre-compact set is pre-ccmpact and the fact that a complete pre-compact set is compact. We note that if A convex space pre-compact in Y , and if X. is a subset of the subspace X A is pre-compact in Y. it of is also the - 27 THEOREM 12.2. complete convex Hausdorff space. in the subset A of the weak closwe of Proof. By lemma is X A is complete, (see is B w(X,X') pre-compact. I t z6 neighbourhoods w ( X,X1) w ( X 1*,X1) [ 4l b . Let for X. u€ on Then X Let B is is B = cl^ A w ( X 1*,X1) is w ( X 1* ,X 1) It suffices to show that We shall show that zis relatively on U° . for each z ^ 6*2 is relatively if and only if for every V w(X,X1) A be a base of closed absolutely convex complete it will follow by Theorem neighbourhood It follows that p.49 Lemma 3) and since X 1* compact. X 1*. B = cl^A . continuous Let X ' or topology on X '* . w ( X 1* ,X1) . w(X'*,X') B C x , for then then ic weakly compact. has thetopology pre-compact X In this theorem, polars are taken in 13*1 X 1* be a If every sequence of points w(X'*,X') pre-compact, as the topology when X has a weak cluster point in A induced on it by the U° Theorem) . Let (Eberlein's . Since that X is z€ X . w(X',X) continuous on e > 0 , there exists a w(X',X) such that V n U° *• I (z,y) I < c •* I (z/e,y | < l . So z is relatively z € e (V H u°) ° forsome Suppose some U° . bourhood w ( X 1,X) continuous on z w(X',X) neighbourhood is not relatively There exists V in neighbourhood X, e >0 V, there exists y € uC n {x f, x 2 , .. ., For every | (z,y) | > E i,i X 1 . V w ( X ',X) continuous on i.e. y £ V H u° For every weak such that {xi,X 2 , .. , x^} such that {xi, ... , x^} C x , there exists and if and only if such that for every weak neigh z £ E (V n u 0 )0 . Thus, for every finite subset exists U° of |(z,y) | > E. X, there \(z,y) | > E . y 6 u°such that |(x^,#)| <E/s , 1 < 1 < r .................. (i) 18 - Now z G B =c*x ,*A, yn > of xgz + e/ 3 { i/i , yit hencefor every X', there exists •••* y j finite subset x G A with • Thus, for every finite subset {yi .ya,• • there exists x«^A Using with |( x - z ^ ^ ('-) and of x>' |< e/s ,l < i < n .................... (li) (ii) , choose y\ G U° such that |(z,yi!>c ? choose xi G A such that | (xi- z,%) | <6) choose y,GU° such that | ( z ,y2) | > G and choose x 2G A such that) (x2 - z , ^ ) I < ^ choose yjGU° such that | (z,y3> | >£ and Proceeding thus, we obtain sequences ? | (xi ,y2) I < e/3 = 1 ' °r 2 ' |(xi,y3' I < i/a , * - 1 or 2 • (xj in A and ln such that, for all n, , |(x -z,y,)| < E/*, . >e and l(xt,yn)| |(z,yn)| U ^3* ................... (11W By hypothesis, 1. * w(X'.X) (xR) has a weak cluster point compact and so to„l Now t has a «e,k cluster point dC 0 . may not asset, that these sequences have convergent subsequences. However, for each numbers (x.y) ,6 X and (x.yj re.E--Uv.ly, and so w exist sequences Urn X^K and each (m^ yex' , th, sequences of complex cluster point, h a v e may claim that for each and (x ,y ) - (x .d) n n> n (t.pl and .x,<l m . n C N , there (nx) of positive integers such that and Urn (xmyiK^ * VI ' * Rl th* of this section, we give an example of a sequence in a convex space which has a cluster point, but no convergent subsequence! . Fw,r each fixed and so X, if then by 11m | (x^ - z »y ) I = |(t-z,y y oo ^ A Also, for each fixed |(t-z,yn x ) | > e - e /3 . iS my > a X , using Thus contradiction of lim (iv) (UD, )| /3 (Hi) . I ^'^nX1 1 . . . . ^'^nX1 ' I ( t , y n X ) I = I(t'd) • > 2C/3 and the theorem is proved. (iv) . - 29 Let COROLLARY. a subset of has a cluster point. I is is If X is complete under ppoof. Every sequence of points in Let be the weak closure of B w(X,X') compact and so 8*2 Corollary 1, B is is also and so then T(XtX') A is A, being a By lemma 13*1, A A of a topological space Thus, i* A is a equence in is A X T pre-compact is said to be has a convergent A has a cluster point. A X, We may conclude is a sequentially compact subset of a convex Hausdorff space (X,T) which is complete under sane stronger topology of the dual pair is T-closed subset, of sequentially compact subset of from the above corollary that if A By Theorem T-compact. . then ever} has a weak cluster complete. sequentially compact if every sequence in subsequenc A A, then, by the theorem, w(X,X1) T-complete. T- complete. A subset of A T-compact. point. B be a convex Hauodorff space and with the property that every sequence of points of X A B (X,t ) (X,X') then the x-closure T-compact. We conclude with two examples which show that compactness does not imply sequential compactness and vice versa. examples are given as an exercise in [ 5] Let let H p.200 be the set of real numbers. Ex. 37. For each CQ - C, the field of complex numbers, and let closed unit disc in n r a 6 *a C .Let IT U , “ d B be cCiB0' induced on it by X. X F otf-lR , be the be the topological product a subset of Let These X under the topo) ogy be the collection of elements-of which have at most countably many non-zero co-ordinates. is a compact subset of that B X by Tychonoff's theorem. is not sequentially compact. it is not compact. F Now B B We shall show is not closed in X, so We shall show that it is sequentially compact. — 30 — Define the sequence for each natural number n ,xtn)) ,x(n)) and each has a convergent subsequence. sequence (t^) q n > N B a € TR. For each x (n) » = eLrJ[a e a Suppose that qE n , the set (eiTrtna ^ is a constant. divisible by Thus n converges > N such that *» t , - t n+1 n is q. We choose a subsequence 2V V2 ... Vn _1 divides (11) > 2nVn (U ) n of ft ) n to satisfy J where for each From (ii) for each n E N nEN. we have that V ^ > 2n *2n ’1 V , > ... n+1 n-1 > 2n -2n'1 ... 2 'Vj = 2l3n(n+1) vt and that u ^ 0Sr(2n+r-l) vn+r > 2 vn , for each n and 00 each of the series J r _1_ nEN, and T r*l lim n -*» — J and r and so ei7TVn a Vn a o " Vn (^ + nEN (i),if + 7 is even is an even integer and cd’d ' = 0. q is convergent for Now let ot ° = ER, e i7TVn a e giTTu^^a , ^ - 1 ^ 0 converges for each real n Vn which 1® of course a convergent series. n-1 Thus, by It follows that n+r and each | + N. r=l V ^ n r=l V n+r 2 1 2 + V 2 + Vs+ Vi,+ ••* • Also, for each in 00 oo each ) {e^71<"nct : nEN } has at values, and so there must exist an integer ■*e (x Choose a strictly increasing 4 (i) by of positive integers such that for each real a. most in Vn a o “ J'n + 1 + Sn ' -*• 0 a. Now,if n + 2+ V n ( L _ + ~ “ + n+1 n+2 V a = k n o n as where + 2+ S n-»« . n is even, ... ) ,where K Similarly, if fn is even. n n From this it is follows that e n o does not converge whichis a contradiction [ Since (x )is a sequence in the compact set cluster point, and so we B, it must have a have an example of a sequence which has a cluster point but no convergent subsequence] . Now let Q/( 1) be any sequence in F. For each j/^n) is non-zero for at most countably many values of a . there are at most countably many values of a Acx C Denote these cx : n ^ 1 '2 »3» •••} by a1'a 2» a 3» ••• of thenatural to a point • is not {o}. C Choose a subsequence (y^ (nk )) .Choose a subsequence z2 in C . the S6que„=e Then, of course, convergel. to . zv ^nk' . zi Proceeding in this way, for each natural number that, as (rV k = l , 2 , ... k -* ® , iv(nk)^ “a Now let r z/E f converges (n2 ) a.l subsequence , k -*• ® . as r, we obtain a of the sequence of natural numbers such c„ r “ a r * j y& q be defined by = z ,r = 1 2 ^ r t- I ••• = 0 We shall construct a subsequence of Consider the subsequence the sequence of natural numbers. (y (n)) if a # for any (nj , n 2 , n^ , ... ) - (n*) To form this sequence, of wehave and then the second term of the second subsequence and so on. r, the sequence ‘nk ’k.l,2.... (n^ n ^ «nd so.'fc, e,=h ...) For is a subsequence of i X ’ . ^ r If .k 01 * otr for any r, then y (\ ] - o for all k. Thus y r. y . which converges to taxen the first term of the first subsequence constructed above each of k k=l,2,3... that P°int values (n1 ) k k=1,2, ... ai °f Thus for which the set numbers, such that the sequence Zj in n, r e f e r e n c e s . 1 . Dunford, N. and Schwartz, J.T. Theory. Linear Operators Part 1 : Intereoienoe PublishersJ Inc. = ss-rss; r . De'kker Inc. (1976). 4. Robertson, A.P. * Robertson, Wendy. Cambridge University Press 5. 6 . Schaefer. H.H. Willard, s. Topological Vector Spaces. (1S64). TopologicalVecSpringer-Verlugd m ) General Topology. M.SC. TOPOLOGICAL VECTOR PROJECT SPACES NO. : C O W ACT t>y Janet M. 2 Helmetedt LINEAR PAPPINGS A notatton Itst and an index of definitions for this project can be found in the candidate 's dissertation INTRODUCTION By "convex space" we mean space over the field If X C and mapping from X "locally convex topological vector rf complex numbers. Y arc convex into Y, then T if there exists a neighbourhood such that T(U) C If X continuous spaces and T is a linear is said to be a co n ta ct lin e a r mapping U in and Y are convex dimensional subspace from spaces and X into Y N, is compact. ! jpology, and its closed unit ball B be defined by such maps will map X We shall ^ % Ihen an(3 let T We need If T T T : X *- Y is continuous and its Y. Any I'near combination of of Y. Y V to of X on theconvex on which be 1-1 and onto V * Xl-T Y.If for X , V(X), and, if this fails, all is V(X) * is not compact, it may happen that distinct from vn+1(X) and B. be concerned with the following problem : the obvious space to consider is X *= 2,1 Hausdorff, any into a finite dimensional subspace X, is there a subspace and so on. Y has the tuclidean N and if T is a compact linear operator invertible ? in is both a neighbourhood and G Y, zq T(x) = (x,yjz^ . X G c K whose image is a finite For image is a one-dimensional subspace of space Y is T, being continuous, maps a neighbourhood into For example, let If and a compact set K . linear map T compact. X V (X) V (X) n.This happens, for is example, if is defined by T(x , x , x , ... ) = (x , x -x , x -x , ... ) 1 Then I-T 2 3 1 2 is the "shift" operator : (I-THXj, x2 , x$ , x^, ... ) It may also happen, if dimensional, that V 1 3 * - (0, Xj, x2 , x3 , ... ) T Is net compact, and is nilpotent We shall show, that if T i.e. (ii) Vn (X) X is infinite for some is a compact operator on then there exists a non-negative integer (i) vn (X) * {ol X n n. X and X#0 , such that (X) ; is the topological direct sum of Vn (X) and (V ) (0) j V restricted to V (X) is a topological Isomorphism and so has a continuous inverse ; (iv) (Vn)1(0) is finite dimensional. on which we can invert We shall also show that topologdcal isomorphism on X V is [ Thus the subspace "large"in a sense ] . V = V, + V 2 and V 2 m-ps Y where X vt is a into a finite dimensional sujspace. r If X v (X) = 0 is infinite dimensional, for any integer We follow linear 1 " T operator (iv) and T as the set of ,11 complex numbers d“ 5 n0t haVe 1 continuous Inverse. X such that This definition can be continuous linear operators on ,1 1 88 * n°n-commutatlve algebra under the operations of composition . Hilbert, space. preclude | 1 | p . 142, In defining the spectrum of a continuous justified by regarding the set of * (ii) r . and It is of course not consistent with normal usage in We shall consider the spectrum of the compact linear operator T on the convex Hausiorff space the number o, X and show that, except possibly all elements of the spectrum are eigenvalues, and that the spectrum is finite or a sequence converging to zero. in the last section of this project we shall consider the transpose, T', of the compact linear map that except possibly for 0. T and T' T , X - , . we shall show. have the same spectrum . shall also prove Schauder's theorem : If X and Y arc Banach spaces whose continuour norm topologies, T and if T is compact if and only if Chapter is , linear map from T' 'uals have their , i„to y, then iB compact . The main source we have used for the above material 1 . ( iJ , VIII . 11 ' »s far as I am aware, Theorem 5.6 is , r,aw result We THE ASCENT AND VECTOR DESCENT OF A LINEAR OPERATOR ON A SPACE . Let and let X be a vector space V~°(0) - {0> ; Then, if n and v"r (O) V a linear operator on * (Vr) ’L(0) X, . is a non-negative integer , v"n (0) D V*n - 1 (0) andV~ 1 (v'n (0): If there exists an integer = V~n - ] (O) . n > O such thac V~n (G) = V~"~^(0) , then V"n ' 2 (O) = V' 1 (v'n™ 1 (0 )) = v" 1 (V™n (0 )) = V “n l (0 ) = V~n (0 ) ; and, inductively, the sequence integer V r (0) ■ V n (0) , (V r (0)) ;i > O all r ^ n . is strictly increasing or there is a least such that the V r (0) are all distinct for and subsequent subspaces are identical with is the case, we say that V has this x = V°(y) O < r < n If this latter n . , V(X) , form a sequence which either decreases strictly or has a least integer O < r < m V r‘(0). finite ascent Similarly, the vector subspaces V 2 (X), ... , Hence, either m > O such that the V r (X) are all distinct for and subsequent subspaces coincide with vm (X) . V has finite descent latter case we say that We shall show that if the In m . ascent and descent of V are finite, then they are equal. lemma and 2 . 1 If is anylinear operator on v r,s are non-negative integerst then (i) v -r(0 )= v"r~9 (0 ) if and only Let x E such that (i)Suppose Vr (X) ^ V 5 vr (x) if and only t / v r (x) + (ii)Vs (x) - vs+r(x) Proof. if the vector space V r (0) * V r (0) . V r (y) = x . Then Hence Thus V r (y) * 0 = x . that v ‘r"S (0) 3 V~r (0) . Let x E n v“s (o) = {o} ; v~s (o) - x . ^(0) . V s (x) = 0 and there exists V rfJ(y) * 0 and so Now suppose x V r (X) ^ V y E v r (0) - {o}. V~r'S (0) , then 3 y (0) = V r We know V r+S(x) « 0 (11) suppose Let vr (X) * V 9 (0) - X Vs (y) e Vs (X) . € v-"( 0 ) . Hence Let Then V (X) C V . we Xnow that y - V, * V, V*(y) - V » ( y J V9* ' (x.CV9 (x> . F V " V ^ = (y,) £ say , since (X) • Now suppose that VS (X) = VS+r(X) . Let x G x . Ihere y € x that VS (x) = vS+r(y) . Now x = x- such Vs (x - Vr (y)) = (>, hence Theorem finite ascent X = ^(X) © n (X) exists vr (y) + Vr (y) . on the vector space m, then (0) = V and m = n , X has and V~n (0) . m < n , then vn (X) , "V the previous lemma, ^(X) V- m (0) V and finite descent Proof (i)Suppose Now x G V S (0) +• V If the linear operator 2.2 ?i = v"(X) ,hence «= v"m_ 1 (0) ,hence (ii) Suppose V (O) and so by (0) = {o} . Vm (X)nv'1 (0) = {o}. By the lemma, m= n . n < m . V ^ X ) = Vn’+ I (X) and so, by the lemma V 1 (X) + v-m(o) = X . But v"n (0) - V- m (0) ,hence Hence V 1 (X) + v"n (0) Also, Vn (X) - Vn 'l‘n (X) . Thus Suppose V By the lemma, Vn (X) = m = n . (ill) v"n (0) *= v"n _ n (0) . If = X. V X * Hence v"(X) ® Hence v"(X) + v'n (0) = X If, conversely, V maps theorem it is 1-1. X ^ X 0 . by the lemma. with finite aecent and d 6 hwv.it. O, so by the theorem it maps onto X , its descent is We shall show in compact operator on a c descend, where X by the lemma V- n (0) . is a linear operator on is 1-1, its ascent is Vn (X)nv'n (0) = {o> 0 X Xl-T X . and so by the the next section that if tcx space, then onto T is a has finite ascent and 3. COMPACT LINEAR MAPPINGS. A compact linear map is continuous. is a neighbourhood in the convex space of tne convex space neighbourhood in number Let X ^ 0 Y, and Y. Since easily seen that and T,S K T»S T X E C . : X -» Y and RoT K into the compact sets compact set K+G , and T X, be there exists a complex . be a compact linear map. It is into . if % Y, then so are T+S into U any ; R : Y -» Z then Xu maps where S : V -» X X From the above remarks, we see that on the convex space Let V map the neighbourhoods C , and T(U)Ck is a compact subset are both compact maps T,S For if K T(Xu)Cv and so are both compact maps from XT , and is bounded, be convex spaces and continuous linear maps and If X is a linear map. XkC V such that V, X, Y, Z T For suppose maps IKV U T+S and into the Xk . T is a compact linear operator then so is any polynomial in T which has no constant term. We shall make this study a little more general by considering"potentially compact"linear maps. if and only If S A linear map S is said to be potentially compact is compact for some integer k > 1 . This general isation does not complicate the work, for : Let Then V V * yl-S and is a factor of V W W. 1c k » u I-S k where S is a compact linear map. Let the other factor be U. Then 1# UeV = VelJ * V I-S « Xl-T operator V = Xl-T, where operators U say. T So, instead of studying the linear is compact, westudy continuous linear and V which are such that UeV = VeU «=Xl-T , where T is compact . — NOTE j( >jid ^ are filters on a convex space 1. a non-zero scalar, then we define > >nd Xjf to be {Xr - F eJF, . is a filter on and X. XhenJ + f Iff., and^.b 3.1 f/ T ta *” (1) v (11) (O) v (lii) „ e, i , , filter base and . then JT t f ;.3 finite dimensional / 18 °PeM aa a map /y*om v(x) (1> ^ * * °°ntimous is closed in % onto Let X F " Th0n d Z n s L T Ct‘ ; X . Let T map the neighbourhood and let N = v (0) . Suppose (rtG , P e > tod a compact Z W a r operator on t%e oonwar aawadbrff i:\’ T. hi (11) to be and X Xa . Theorem _ t * X U " T'*' Into the compact set •Hence haVl"g a — Xx£ T(ul C t Iu ) is not open as a map from W £ -W" such that W C U Let and V(W) B - 2„n C x. and so neighbourhood. Is finite Let H r be a base of absolutely convex neighbourhoods In v K , X onto v(x) . x . Then there exist, Is not a neighbourhood In v(X) . . We ,bow firstly that the sets whe » ( " W form a base for a filter on Let h c V(W) and x , W. ifn o v(xl does not meet V(K) is a neighbourhood In C(V(W))0 V (x) then v(x) , which Is a conrrc contradiction. Let x £ * n v(x , n snd let we show thatthere exists b , e a, n = w nv-hA,. Since w the form It y u , , - V,y, . 0 < u< 1 L 2h , let u » or ,0„ , with = < , < 1 . y, „ . , such that 1 . is absolutely convex and absorbent, the set IO.a, then Supposey , 2 * . tX , X y £ w ) „e„ce the set Is of (X: X y £ 2 w ) - 7 - is of the form [ 0,2a) p y <= 2w n cw n v ' 1 or ( 0,2a) . Choose (A) . We have thus shown that the sets for all (v"1 (Ai) O A ),A ^ X 1 V ‘ (A) Ci p with T(B)Gt(^1) at some point z , = and 2K . and so 0 . V(^ ) -*■ 0 , Hence V(z) map from X onto V(X) v(X) .By (11), V(U) V(b) G ( a + i W ) ^V(X) 2 i a G v(X) , It follows that the sets , Hence these sers V(3*) > a .Also, V W+N is open is a neighbourhood in Since a G v(X) , V(X) . the set . T( (b + U) n « X (Vj() Thus Xa wG" 6 / are non-empty, (a + -^-W ^)Clv(X) C v(b t U) , it are also one meets every other. also applies to the sets V 1 (a+ W)t'Mb + U) , form t±e base ofa filter «X on v _1(a + W ^ ) G t (j<) It follows that , X. hence = T(3«) and so t (J") . Hence T(#) + U»V(j^) clusters V(y + U(a)) G V(X) X a G V(X) for Now, T(b + U) G which is compact. A lsc, V(X3^ clusters at . W ^ V(X) . 1 (a+W)Hv(X) , with clusters at a point y G T (b) + K . is closed. and then (a+^-W ) n V (X)C v (b)+W '~iV(X)Cv (b)+V (U) =V(b+U) . T(b + U) * T(b) + T(U) C T(b) + K , V(X#, , B , (a + W)^V(X) ^V(b+U) - (a + W) O v(b + U) The above state..ient y + U(a) clusters j- z G n <^B . Thus 2 the sets non-empty, and clearly each at zG n . V(z) is also small of order and each one meets every other.Since Now X$ Hence 1 ) n V(X)is non-empty. W G "6 /" . clusters It follows that G 'tf such that W ^ V(X) C v(U) . Now, since • Now . 1 (a + ~W T(jf) j z , but ,by definition of as a W So ■+ 0 . clusters at = 0 and not meet B , which is a contradiction. Let a G U»V( ) Now Kjf) = U o V ( $ ) + t ( ^ ) does Choose hence T(B)C T(2W) C 2K . W+N is a neighbourhood of (iii) A^ and A €"<</' form the base for a filter & z € Xp" = Xi" . Also, V(X^r) X(V(S*»)) A^- A ■i*)' with A Now Let (A) Cl p are non-empty n B) d v' 1 (a3) n p . which contains B .Also, , hence V(X^r) there exists H (V™1 (A2 ) B) So the sets at V A E "6 % Also, if on U e (a,2a) , then , and a G V(X) and V(X) M N M t f - 8 - If, t% addition to the hypotheses of the theorem, COROLLARY. VtV-VJJ , then for any integer dimensional and r >1 t is on open map of v ' (0) % is finite onto the oloeed suhsvaoc . ^(X) Proof. Ur. V r = (UoV)r = XrI-S where S is a polynomial in T without a constant term and is thus compact. The corollary now follows from the theorem . lemma let 3.2 be alinear operation T which maps the neighbourhood w - Xl-T with in where p n (2 u) vith Proof. V Then there We first show thatF ^ G + U X u C G + b c G +e c C C G + e [ G +T(U)] G +E T(U) G +E K . K is a . X(UHf ) exists a voint For suppose FCfi + U . [ sin e G C F 1 There exists V > 0 such that bounded set.For every positive in particular , F C G + V . £ , We now show that We shall deduce that F n 2% F n 2U 4 G + U . F n 2nU C Let 2% e wh,r« Let 2nu - g + 2n" 1 u i . p n 2n u C G t F n U above. exists 2"‘l u C n. 3.3 G + U ..... F n 2U r G + U . n is any integer G F , ' 1 as GCf. integer n ' 1 . G > G > r n 2n ": f E F It follows u - g . f Is contained In that n 2n"2 u c ...c 2n 0 for some W. may now conclude that F n 2u 4 G f U , and so there , let T x*fG + u * be a compact linear operator on the convex % , If* linear operators such that U and V > Hence F C G . u . which contradict. Wawaddrff apcee Then V V. It follows that x € F n 2U Theorem g J- e 2nu » 2n"^(2u) € 2n-1«* + U) * G + 2n-1U for every is Absorbent, each positive Integer ,1 , 2“ where C K C ViV f F C G - G . H Suppose that G + F n 2n - 1 -J uEu. Then F n 2nU c G + 2 n~ 1 U n F , F From this it follows that ■Ihis contradicts the hypotheses of the theorem, and so As x ( W(UOF) + T(uriF)] be anyneighbourhood. since G he distinct subspaces of X x?G + u . Let e > O . F e e X f C g +€ Let F and and W(F)C g . G C f x U into a compact set K • let \ * o , and let g closed, on the convex space and let 2* two continuous U» V - V »U "= XT- m . have finite ascent and descent . gf Proot Let T m p the neighbourhood U Into the compect set X . ToA. M e Ue V * Xl-T . end r # s - h r * N s . „«„=« v'(W,x„ Let - u.v^,„ We now apply Lemma e „ch r > 1 If , * * V l - = - 3.2 then ^ s o with V (x «,x, e h , - F - l,r+i # ’ there exists , point -= • »us w .h ^ , c h , r x, e . with ,,4 & * r > s , »<V Tlx ) - T(X ) - X,r - «(xt) - Xx, * Xxr + N r , G X x r - N r - N r + Nr = as N s C N r+1 and W(Nr+1)c N r • Suppose that for some T(x ) - T(x ) € X U . Since r T(x^ r a n d s with r >» s with r > s , - ?(*,) C X xr ' N r ' X x % T ( x ^ - T ( x , ) - ^ C X U + ^ . for every and ^contradicts (i),sot,.t , A \ T(xr) - T(xg) f X U . we shall show that this c o n t r a d i c t s the fact that There exist ^ How, by If ,11, . T«.r, * T „ „ , Also, x, E Ny+i " (2U) r,s,l with r > . T(x ) - T(x ) E XU r 8 £lnltes“ ^ ' v , a t „ . „f,x, and so such that . V 2X 1- such that 2K C ^ s .so the points T(x,)C 2K T(x,) E 2X __!= pre-compact. for each + Xu and ly, t X„, . Tlx,, are all c.stlnc, r . T(x,) r + >U This i, a contradiction, and so V must have doesnot have finite descent. are closed (by 3.1 T h e n Corollary, and distinct. the sub.p.ces By Lemma with 3.2 , for each r , z r G Mr n 2U If there exists a point r.r ^ Mr+] + U with and T(z^) z^. € m 2K . 6 r < s , T(zr) - T(zs) = Xzr - W(zr) - Xzs + W(zg) G Xz^ + M r + This leads to a similar contradiction and so COROLLARY If 1 . direct sum of n V n (0) Proof By Theorem is the ascent of and V V , . 1 has finite descent . then X is the topological \^(X) . 2.2 , the ascent and descent of n and X is the algebraic direct sum of V n (X) are both V n (0) . corollary to theorem 2.1 v"(X) is closed. is the topological direct sum of a n d v n (0) , COROLLARY Thus using If 2 . T (ii) (iii) (iv) W = Xi-T , is Use theorem 3.4 Let maps 3.3 X W X * 0 of , the following are equivalent : T ; X is 0; is 0 ; itself . with T, Ut V be U «= W , V * I . linear operators on the with T compact and v u = Xi-T, where U-'V = (ii) W X onto space (i) of the descent of W X * U, 0 convexHausdorff V continuous, satisfying Then . is the topological direct sum of the closed subspaces l/^iX) = M and V(M) » ond M V n (0) = H where n is the ascent of V ; V(N) » N ; (iii) V is a topological isomorhphism on (iv) N is finite dimensional and \*(fl) * 0 ; (v) Vn (X) ; 1 - 1 the ascent (v) is finite dimensional and [ 1 | page 96, Prop. 29 Cor. . Xis not an eigenvalue w By t8 a compact linear mapping of the convex Hausdorff space into itself and (i) X V~n (0) and V t h e Theorem ^ 2U For each integerr > 0 , V (0) V ; and X / ^(X) have the same dimension ; (vi) V « V + V 1 where V is a topological isomorphism of 2 onto itself and V^(X) is finite dimensional. X - 11 - Proof. (i) (ii) This follows from 3.1 V(Ml = Vn+ 1 (X) = M ; Corollary and 3.3 Corollary (i) . and V(N) = V(V-n(0 )) = V(v'n- 1 (0 )) « VoV~ 1 (N)C N . (iii) U(M) -U(Vn (X)) = vn (U(X)) C Vn (X) it follows that operator on U,V,T is on (iv) M . m tl.at 1-1 V on . Clearly M . Since Since 3.1 with Hence V linear X=M and V(M) = M , it follows is an open mapping . M . XM = M , T is a compact We may thus apply restricted to from 3.1 V T(M)C M. By 3.3 Corollary 2 , is a topological isomorphism M . Let r be a positive inteaer, then X/Vr (X) * (M+N)/(Vr (K) + V r (N)) a (M+N)/fM+Vr (N); . wv show that (M+N) / (M+Vr (N))3! N/Vr (N) . Let Let n I m+n + M + v r (N) = n + M + vr (N) G (M+N)/ (M+v (N)) . f(n +M + M + V r (N)) + V r (N) * n and since 2 = n + Vr (N) , + M + V r (N) , M Cin ” {o} , n 1 - n 2 then then n € v 1*(N) is well defined, for if f 1 - n 2 and so + Vr (N) , n + v (N) ■ n l It is easily seen that f Now dim M ■ dim V r (N) + dim (N/Vr (N)) V r v0) C N hence is linear, 1-1 € M dim N = dim v"r (Q) + dim (N/V_r(0)) . dim (N/vr (N)) = dim V~r (0) (vi) Let P and and maps X/V But - dim X/Vr (X) Q (X) 2 + v (N) . ontoN/V (N), ...(*) and N/(Vr)" 1 (0) a V r (N) , by hence (*) . be the projections of X onto M and N respectively. Let of X V onto itself ? ■ VoP + Q , then is a topological isomorphism for (a) is clearly linear ; Let (b) V x «= m + n E x suppose m O n M , m ^ M (x) = 0 = V(m) + n . •= {o} hence m ” 0. WmBM where , nG N .Then Now V(ra) r M V(m) = n = 0 . Since Hence is 1-1 . I V and is 1-1 on (c) since v is an isomorphism on such that maps The VtnU x = m . onto M, there exists Then V (m + n) = m + n The continuity offollows from that of (e) V'1 (x)- (v|M)-1 (c) . NOWlet V, - v < corollary space (P-x)) * 5<x) , from follows from that - 6 , then V^x) Let * y * ° : the of V iri the above theorem. Proof Let S be a polynomial with * of V,P P,g and and and continuous compact . Then if tee mpvZ-S *<£> be the quotient when By the remainder theorem, . v_ t V;. v.(p*g, . v be a t,(S) Q (v Im )'1 . , »(„, - n e N . ,„d Let 1. X . € M X . <d) continulty of V ^ 1 m *(() lMC> is divided by (u-V . finite asce (J-u) _*((, and so _ Ip (S) e(yl-S) = 6(U)I - ,(S) . We now apply theorem A = $(y) , 2‘ 1,8 a conpact linear operator on the convex Hausdorff and WI - l-T , X * 0 , In addition on not ia Proof. o Let I - > M an w and S . commutes with eigenvalue Then s M . Also T.,XI.S , , ( X l - w ) . x V 1 - l2 IX„ - l - „ hence s is compact. Now suppose Then Xl-T COROLLARY maps If 3. space eigenvalues eigenvectore. X * 0 X X , and onto T U X X w"1. X"2 ,Xl - is 'a \ * o , m and u'1 has . x2 , on M . x -2 TiS . W -1>K . and so Tat . s .T .-xs . is not aneigenvalue and If T Iscontinuous . t - X -1T - X -1S a X"2 s . T T . of - (Xl-T) .X-2 ,X,-S1 . I - X-1T . x-ls . X~2 {Xl-S)a (Xl-T) an which then of theX ^ d i - s , we restrict all mappings to ■I M - ^fx) is compact linear operator on x , u - *(S) 1 tn the above theorem. topologicalt ,f'7 h p r m o s i withV - yi-s = <j)(S) . X space 3.4 t Te(Xl-S) is of . compact , T . M - X . is a compact linear operator on the convex Hausdor” then only. apart possibly Each eigenvalue from 0 ,af has a Proof. X Let (i) % * , XE 0 Xl-T (Ul) T . epectro. of is not Thee .Ither 1-1 ' 'u-t’T x - L end -aps X onto XtseXf. tut Xos Xover.e is not continuous . Now (iii) is impossible by Corollary 2 and equivalent by 3.3 (i) Corollary 2 . , o u °“ s £rom finite dimensional , where and CTE all SPECTRUM — v 1<o> x • r E B r . : " t ,e Spt of 4 fact u>at Xl T then the equation in «t>(X) * 0 1 -— V ^ OP A 'U-TH-O euoX * COMPACT LINEAR OPERATOR z : : r: xr» a finite act or o f a sequence coniwrpent to stfv . We have seen that, apart possibly from 0, the points of P: " T . ™ “ := ZZZr~~ (X ) of distinct eigenvaXues of J iwlth ‘ r' b. a -equanoe of corresponding a i g a n w o t o r s , and Let W r - XrI-T I r> 1 • Wr(Hr) ” Wr < S '*%' ' ' . <»r(x r. 1 2 - <Xrxj - T(x^, « <X x - X r i * H -l »„ Xr ....... " Xrxt - ?(%,)....... XrXr-l c , X x ii r « V X , > ' ’ lnCe r 2 - X x , 2 ... Xrxr-i ' >r-r r j plnce none of the above elements are ? ' Let T map the absolutely convex neighbourhood We apply Lemma 3.2 ere exists a point Then If T(y^) 6 2 with W = yr F " Hr , , such that yr ^ into the compact set For each G Hr " U K r > 2 * r-1 ' U 2 and yr (i) "r-l^ K . r > s , (ii) T(yr) - T(ys) - Xry r - W r (yr) - xsy g + Suppose that T(y^) - T ( y J G e U . ^ XiY r + Hr-1 Then from Xry r G T(yr) - T(ys) + B r Hence U + H yr G C u + H Xr x » _ 1 , c e u + Hr since _ 1 . U is absolutely convex and r tol, contradicts (1). hsncs Tty,) - Tty,) * € 0 , By the same argument as that used In theorem Of the fact that :K X for all r.s with r > s . 3.3 , we obtain a contradiction is pre-compact. Thus, for each positive integer eigenvalues (ID such that n, there exists at most a finite set of |X| > £ . either finite or countable. Hence the set of eigenvalues of T is Clearly if the set of eigenvalues is infinite, it is a sequence converging to zero. COROLLARY ,paae let % and Let S be 4 ccnttnwoue linear operator on the oom'ar Hawedcr/y $ b* a polynomial Then the speotrwm of zeros of S K S J compart . <t> . are non-zero, then S , X ^ 0 , 4)(X)I - with is finite or countable and its only limit points are Proof we first show : of fnot a constant; 4> (S) . <MX)?. - 4>(S) . if *(X) 4>(X) If If in these cases, since X is in the spect-um of is an eigenvalue of * 0 *(S) and let the polynomial Xl-S or Xl-S or *(S) it'(S) do not map $(S)is not . i|> X S, where X G Let satisfy onto 1-1, nor is > and spectrum ^(S).(Xl-S) X , nor does *(X)I " is a compact operator, <t>(X) • is an eigenvalue of <MS) • If Xl-S and IMS) are both suppose that the inverse of I-S $(S).(*(X)I - M S ) ) ' 1 , $(S) and 1-1 and both map Thus, in all cases, *(X) X is not continuous .But onto X ^ (Xl-S) being compact, tne right-hand siae is continuous, which is a contradiction • is an eigenvalue of *(S) . -MX) If the spectrum of S were uncountable, since a polynomial In a lomplex variable has only a finite number of " r o e s . It would follow that the set of eigenvalues of of S »(S> X be a limit sequence (X^l point of the spectrum of S. <MXn> ♦ *(Xo :, and, = in - m a n y distinct values. The * Hence is a polynomial d l X J . O . *(X,I, above corollary is al, ■ true If bourhood and so be finite dime X a has Infinitely is a zero of < is a constant , a compact neigh S a finite set. THEORY If apace X , its transpose, T ’ is coxpact when •jhere A T is a compact Let compact set has the topology - T map theabsolutely convex neighbourhood Also, using and hence V . (T(U))° = (TUf) HI P-39, lemma Into the . ) 0 , 6 Then V V - (T1) is a neighbourhood in X' . (U ) T* (V) c u Shall show that U° is compact . if A is the collection of all closed x , U K . Let then U° is -9 , A °C Now operator note X' lireav is theset of absolutely convex compact subsets of ■ ■ Proof. hence then 0.1 . absolutely convex pre-compact subsets U° is also compact, and T' is a compact (SWT) ' « S S.T + T* I 1 are continuous linear operators on (Xs) ' - XS' I ,S-T > ' *' and we assume these results. 5.2 % , euoh that 1 . theorem on X' . X e c Theorem By Project compact . It is easy to show that if I* ■ I » space * .lonal and the spectrum of 5.1 and and would have LEMMA of exists - MUJ! and so Xc SfO then the identity map wool,,be compact. DUALITY There of d i r e ct paints of the spectrum such that Now we Hence the spectrum is countable . Let for would be uncountable. Let T % * 0 , a compact linear operator on the conoez .%i<r -err; and % and V*V - V.U - AJ-T . fij y and hat* ? continuous linear operators on A Then some ffirite; ascent and descent (ii) for each positive integer r , '' r (0) and (V) (0) n ; X (iii) if X ' has any topology for which then X and is compact , is the topological direct sum, of (V’)~K (0) V' T' (V) (X') . On the finitedimensional space is nilpotent, and on (V')n (X) , V ( V ) ^ (0)t is a topological isomorphism . Proof lemma (i) if r (Vr (X)>° 6 - ((vr)'f 1 If and so ascent o'; is eny non-negative integer, then by [ 1] page 39, V r (X) V = V r+ 1 (X) < descent of (V) ~r (O) - (V,)_r_i(0) V < ascent of V V . ( V ) ' r (0) = (V')'r-1 k0) Now V (X) then is closed and V r (X) = V r+ 1 (X) absolutely have the topology e&° where <>&■ is the set X" convex compact subsets of and so the transpose of X . T* is ByLemma and X* The result (i) (11) where the 5.1 , T* X1 is X , T1 , V and T . We may thus reverse the roles of , X Thus, and so der- ent of is compact, and by che Mackey-Arens theorem, the dual of V , . Now let of then . (Vr (X))°c = v r (X) = (( V ) r (0))c . absolutely convex , hence if (X°) = (V’)"r (0) T and , to obtain the fact that ascent of V= descent of V . now follows . We have that polar is taken in But (Vr (X))° = (V1)~r (0) v"r (0) = X/Vr (X) X1 , hence . Also, X/Vr (X) « (Vr (X) and we have used [1 ] p.78, proposition V~r (O) and (V) r (0) have the same dimension . Now (iii) follows from theorem Apart possibly from corollary eigenvalues . V = XI-T . Let X be a if y Also, the equation in x r, vanishes on 0 , T ard T' have the same non-sero eigenvalue of T Then the equation in only vanishes on (W) and f and let x , wix) = z/ hasa solution if and 1(0) . w 1(x1) = y ' has asolution if and only if y r w ^ (0) . Proof Let ascent of X = 3.4 . X ^ 0 If X V = 0 = ascent of Now exchange the roles of T is not an eigenvalue of V , and X and T1. T , then the is not an eigenvalue of T1. - 17 (W(X)) = (W1) ^(0) W(X) = ( (W) ( W ) 1 (O) , W(X, is closed and absolutely convex, 1 (O) )° = {z G x : (z,t) = 0 is a subspace of We thus have X' t G w ,'1 (0) } for all since . s W(x) = y the equation y And since vanishes on (W1) 1 (o) has a solutlon<=> y G w ( X ) ^ > y E {(W') " 1 (O) J ° < ^ . The last result may be obtained by giving X' the topology o& ° as in the theorem . We shall obtain a few results about compact operators on normed linear spaces. We note that, if X is a compact linear operator from and X Y are convex spaces into Y then into compact sets, for every bounded subset bourhood which T X [ 2] T then since section maps bounded X has a bounded 55 , Taylor gives the following definition of a T from the normed space Xto the normed space is compact if, for each bounded sequence (T (xn )) T is a compact operator. compact linear operator "T Y, maps bounded sets Y . is a normed linear space, and into compact subsets of neighbourhood, In X T X is absorbed by the neigh maps into a compact subset of Conversely, if subsets of of T and contains (x^) in Y : X , the sequence a subsequence converging to some limit in Y." We shall show that the above definition definition in the case when X is a isequivalent to our normed linear space and Y is a convex metric space. Suppose metric d. X Let is a normed linear space T each bounded sequence be a linear operator (x^) in X, (Tx^) and from Y a convex metric space X into Y with such that, for ct tains a subsequence converging to z e r o . Let B be the closed unit ball in We shall show that T(B) compact . Let (y^) be a sequence in be a sequence which converges to Consider the matrix T(B ) For each yf . : \ T( x i2 ) T(X 2 2 ) T(xmi) T(xm&) .. y n, let (T(x nr )) is We form a new matrix from in the mth row of from y as follows : A, for each is less than m A obtr' 'ng the matrix - . n, choose an element whose distance Call this element T(U^) , tnus B , where 1 T ( uh) .A T(ui 2 ) T(U 2 l) T(U 2 2 ) T(umi) T(um 2 ) B * For each n , d(T(unn) , Y n )) < on the diagonal of sequence T(V ) B has a convergent subsequence. and call each corresponding Let the subsequence appears Let £ > 0 be given . Let There exists an integer Let Now the sequence n R = max(K.M) , N then Hence every sequence in M T(V Call this sub in the rows in which ) -*■ y € y . be an integer such that such that r > R t Tb nn yn T(u^^) n > N 2~ < C . d('r(vnn^ ' y) < M d(z^,y) < d(zr> ? ( V ^ ) ) + d(T(v^l , y) < 7 + M < e * ___ ) has a convergent subsequence and so T(B) Y be convex Hauedorff spaces and is compact . Theorem 5.3 Let X and weakly continuous linear map from polar topology Jb°, T X into the elements of & maps the sets of X . Let a X ' have the being subsets of X . into pre-compact sets if and only if rnaps equi-contir>uouB sets into compact sets . Proof. ^ ^ "A T" Li Let T be the given topology on continuous subsets of Note that T 7 Y' . Y. Then maps the sets of maps the sets of Let t 4 o T = c- . be the collection of equi- intopre-compact sets onto pre-compact sets, as a subset: ot a pre-compact set ir pre-compact. By if [1] page and only if Suppose E e t. Eoo T'(E) T Then is E C E0 0 Now for each , and E° is a neighbourhood ir T'(E°0 ) JXlaoglu Theorem B a n a c h is W(X',X) T'(E0 0 ) . pre-compact, hence so is Thus h"(T1\ eTT" is Thus Let X T*maps E operator free, X and into T ’ is weakly . hT'C) isalso^ - (T'(h(E))CT.(h(E)). into a compact set . (i) • be no-med linear epaaes and y T a ccnpaat linear f it a U o a ccnpaet operator when Then r . complete. pre-compact and complete, and hence it is The converse is clear from COROLLARY 1 and hence compact and complete . i= compact . Let v p . l 0 5 , p , o p o , i t i o n 3 , Corollary, l ' ( E - ) T'C) ^ z < 'b . sets of ^ into pre-compact sets. W ( Y ,,Y) compact by the [11, 3 , T(D)is pre-compact for each D is pre-compact maps tne continuous and so By 51 theorem X have their norm topologies. proof mis follow, from the above lemma by choosing » collection of balls In sets ana that T corollary Let 2 X, by noting that balls in compact => T X and a linear operator f r m Then topologies. proof Y continuous = > T are equl-contlnuous weakly continuous . be n o r m d linear epaaes, I into X. 7 ’ compactor in the lemma, replace collection of balls in V Let to be the X ’ and X X' reflexive and T have thetr norm is compact . X. Y. T by X'.Y'.T' and let» be the X T' Y' Suppose is T' is a compact operator. W,X',X) - W(Y',Y") maps balls In Y" into compact sets in Then continuous and so Into compact sets In X , but. 1 |v - 1 . T' is continuous and so T" ( Y " ) C X X. Hence T" |Y T' By the lemma, maps balls in Theorem duals T X and be convex Hausdorff spaces 'jhcne Y Y ’ have the strong topologies, X ’ and and let le t Let 5.4 be a weakly continuous linear mapping from Y be am pZeteord kwreZ&ML oompoot eete if and onZy if 3" ^ T Suppose Y hence Y' . is a neighbourhood in and so Then Y . is Conversely, suppose t ll is S(Y',Y) of X , T h(T(A)') Y . ? mrpa bounded eete is Hence Let A W(Y',Y) A0 0 Y is barrelled, be an .AY ,Y) bounded and A° is an equi-continuous set A . T 1 maps T* A intoa compact set . maps bounded sets into compact sets. p.71, lemma 1, Corollary, every equi-continuous subset of bounded, hence T' A Y' maps equi-continuous sets into compact maps bounded subsets of Let Hence A . By the previous Lemma with * sets. into bae tAg same property. S(Y ,Y') By the previous Lemma, By X 5(Y',Y) , /r°* s(x' ,x) has the topology Y' , TW maps bounded sets into compact sets. bounded subset of in 5 ( X ’,X) , the collection of bounded subsets X , onto pre-compact sets . be a bounded subset of is pre-compact, and since h(T(A))' is compact and so T y X, then T(A) is pre-compact, is complete, it is also complete. maps bounded sets into compact sets. - 21 Theorem 5.5 X' Y' and from X (Schauder) Let X and be Banach spaces and let Y have their norm topologies. Y . into Proof Then T Let be a linear mapping is compact if and only if We note firstly, that if T T' is compact. T is a compact operator it is continuous and hence weakly continuous. weakly continuous and hence T If T* is compact, it is is weakly continuous. We see that this theorem follows from the previous theorem by noting that the balls in NOTE: If X X and Y1 are bounded sets and neighbourhoods. the collection of uqui-continuous subsets of If A closure of U in X and & is a convex Hausuorff space with topology f; , X ' , then o # C- is an equi-continuous subset of h(A) X' is £ . then the W(X',X) is also equi-continuous , for there exists a neighbourhood such that A C u° , which is W ( X ',X) closed and absolutely convex . It follows that, if is the collect jn of closed absolutely convex equi-continuous subsets of X' then u = £, We shall use this result in the following lemma . lemma that 5.6 Suppose that X and X ’ has the polar topology absolutely convex subsets of mapping from (i) T X into Y . Y are convex Hausdorff spacest , where Jfr is a collection of X and that T is a weakly continuous Then maps the sets of & intoW(Y,Y') compact sets if and only if T" (X" ) c y ; (ii) only if T' maps equi-continuous sets intoW(X'tX " )-compact sets if and T ” (X" ) C y . Proof X X’ TI + Y X" T' I T" Y' Y ’* In this theorem, polars of subsets respectively. Polars of subsets of ot X or Y are taken in X 1 ,Y’ are taken in X *,Y 1 X " , Y" respectively. Since T is weakly continuous, Lemma 10.1 , pi T" (D00) C (T(D) Now T" (X" ) X" T'(Y')C x ' . The conditions of act 1 , are satisfied and so, if ) 0 0 D €^ , then T" (D . =-= U D 0 0 DE&" , and so " (D0°) C^U(T(D))0° . . . . (a) - 22 - (i) Let D € •8 ’ . B C y C Y1 * . topology on Y , is the Suppose The B By (a), Conversely, suppose W(Y,Y') continuous. and so compact. T 1 D ) ..s into a T(D) Y'* , since induces the D W(Y,Y') Now ^T(D))C ° is absolutely convex and so C y . T" (X") Then D G •6 ' , then D° W(X" ,X') compact. D C D JO W ( Y ,Y '>-compact set W(Y'* , Y ')-compact. T " (X") C y . Let We have that maps (1 0°° D topology on is also closure of (T(0) )°C C B C Y . X1 maps W ( Y 1 * , Y ') and so w ( Y '* , Y') T T" is a of Hence hence is W(X ,X)- neighbourhood in T"f >0 0 } is W(Y,Y') T(D) - T"(C) C T" W(Y,Y') compact set. « T " ( X " ) C y . Then T* is and so Into a suppose A W(Y',Y) - W(X' , X » ) continuous. (b) We apply Theorem of the theorem by W ( Y 1 ,Y) and Y under the given topology, W (Y' ,Y ) Y. some A6 C is W (Y' , Y) so B° is also There exists D0 0 C a neighbourhood into a x" and D0 0 , B By (b) , A G <76 and so . (Y ,Y') T(X" ,X') is a T(X" Also, (c) . ,X") » compact, replaced by • T " (2) C y + A is a topology (X" ,X')) 'T'(A))° . z in X" Hence there exists ofthe . . . (c) , is included « (W(X',X" ) closure of T'(A))° . , X ') neighbourhood x(X" ,X'5 neighbourhood of by and £ is absolutely convexand , by hypothesis in aW ( X ' , X " ) compact set . D W(X' c is absolutely convex) dual pair ) 0 is B _ z G D ' for some D Gj9. closure of D (since closure of such that A C b We need only show that * T(X" ,X' ) is a Y eor We apply Theorem 5.1 of Project 1 with the D (T 1 (A) and maps equi-continuous sets into W(x',X" )-compact closure of D (since Hence Then C C A in T'(B°) = W (X" ,X •) T'(A) W(Y',Y) W(X',X" ) compact set . T' be as above. «=U D G .6 Now is the given . W(Y',V) compact. Let z G X" Y) compact by the Banach-Al Ao^.u Theorem , and maps Let A ° (polars taken in dual pair (X,Y) replaced by the dual pair A Then Y = n (y + a ) . A G /ii Now X under the topology 3 of the theorem, thetopologies A G A Conversely, suppose sets * Y' be an equi-continuous subset of Y'. «/< . T* Then ^ By Corollary B° and so Y' . coincide on each Let Y by Replace the by the collection i/fe of the closed absolute, y convex equi continuous subsets of topology on 6.2 Corollary 3 of Project 1 . , in X " . and z (the x G D , Now z - (T'(A)) r(X" ,X ^ x e z - (I 1 T" (X") c Y . _ _ - t Y gyj y in t 0 y . le t ( r y„, , fx'.X'N , (iii) be Btinaob space, whose ^ Zs X ‘ be the transpose o f T" the Th«n follow ing T" (X") c Y . and since Y is complete Y = ^ • The theorem then follows. spaces ...a T is . continuous linear mapping = J * UnlqUe C O n U n U °US “ t TT . r r (X« x'*) r then ' ^ " " " Z e Z u Zsposn'of T* that 9 ? ^ ^ ,h w that was defined as follows , with respect to the system i f . - • T - T'*|x . mapping . Proof , rl ,t X T ' A /> 1T :::r:nr..r: . ronvex. then B is the «(Y,Y') closure of B . Let T map the neighbourhood U From project 1 lemma Let h(K) - J , 5.6 , then and hence complete. j0 0 V0 0 subsets of Then is pre-compact in be a neighbourhood in X (a x) , which are small of ax - bx (bx) G V ( K X and J = J in Y and Y . We now show that each let T(U ) c (T(U)) . In order to prove the theorem we need only show that is pre-compact in Let J u € % into the compact set K , where A, order There exist A £, A 2,.. V A^_ which a - b G v . , , Then such thatJ c u is small of order be nets in and so Y . V . converge to Let a It follows that a,b € and J0° b and respectively is pre-compact and the theorem is proved. REFERENCES, 1. Robertson, A.P. and Robertson, Wendy. Cambridge University Press. (1964) 2. Taylor, Angus E . , INTRODUCTION John Wiley and Sons, Inc. (19 58) TO TOPOLOGICAL VECTOR FfACLT FUNCTIONAL ANALYSIS. Author Helmstedt J M Name of thesis Closed graph theorems for locally convex topological vector spaces 1978 PUBLISHER: University of the Witwatersrand, Johannesburg ©2013 LEGAL NOTICES: Copyright Notice: All materials on the U n i v e r s i t y o f t h e W i t w a t e r s r a n d , J o h a n n e s b u r g L i b r a r y website are protected by South African copyright law and may not be distributed, transmitted, displayed, or otherwise published in any format, without the prior written permission of the copyright owner. Disclaimer and Terms of Use: Provided that you maintain all copyright and other notices contained therein, you may download material (one machine readable copy and one print copy per page) for your personal and/or educational non-commercial use only. The University o f the W itw atersrand, Johannesburg, is not responsible for any errors or omissions and excludes any and all liability for any errors in or omissions from the information on the Library website.