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C AHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES M ICHAEL BARR Duality of vector spaces Cahiers de topologie et géométrie différentielle catégoriques, tome 17, no 1 (1976), p. 3-14. <http://www.numdam.org/item?id=CTGDC_1976__17_1_3_0> © Andrée C. Ehresmann et les auteurs, 1976, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Vol. XVII-1 (1976) CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE DUALITY OF VECTOR SPACES * by Michael BARR duality theory for vector spaces by introducing a topology into the dual is old - it apparently goes back to Lefschetz ([3], pp. 78-83) - but is surprisingly little known. Briefly, if The notion that V is an many more infinite dimensional functionals less those that continuous ground on field space and V* its dual, then there are are induced topologized by pointwise convergence, the discrete. we over a spaces V* than nice a vector V* when it is being get by elements of V . Nonetheidentified are precisely the functionals which are on be can In this paper vector one can extend this duality discrete field. The to the category of structure involved topological turns out to be rather well-behaved. subsequent paper(1) I will extend many of these ideas to the category of (real or complex) Banach spaces. In a few cases in the present paper, the desire to smooth out this extension has led to a slightly more cumbersome exposition. In a In accordance with the doctrine of don’t itch>), we leave aside all questions Reyes («Don’t scratch if you of coherence. The concreteness of the constructions guarantees that all necessary coherence conditions are satisfied. * N.D. R. Cet article donnees par M. BARR conferences logie et ont et le suivant aux Journees sont le same Chantilly o des deux conférences ( Septembre 1975 ). Ces ete r6sum6es dans le volume XVI - 4 ( 1975 ) des «Cahiers de Topo- Géométrie Differentielle». ( 1) In this développement T.A.C. de issue. 3 I would like to Council of Canada for thank the Canada Council and the National Research supporting this work. In addition I would like to thank University and the Forschungsinstitut f. Mathematik, E.T.H., Zurich, for providing stimulating environments for carrying it out, in particular the former for the use of their magnificient library the Mathematisk Institut of Aarhus facilities. 1. Preliminaries. Let K be ( i. e. discrete) field. By abstract an separated linearly a equipped with a topotopologized K-vector space logy for which addition V X V - V and scalar multiplication K X V - V are continuous. In addition we require that for any v # 0 , there be an open subwe mean a vector U space veU. with By a linear map. The category understood. The described ( linear ) functional is denoted joint V - ) on If V where W is the E B, a linear if the family ( see [1] , [3] , following analogues. same vector a if K is whose The are’the discrete spa- inclusion D -> B space with the discrete subset A C V is called subspace objects a linear of V . We say that V has a left ad- topology. subvariety if A v + W is linearly compact (LC) = of closed linear subvarieties has the finite intersection property for details assertions PROPOSITION 1. are as well as easily proved proofs of the assertions in ways similar to below). The their topological A closed LC space is LC; the subspace and a (separated) quotient space o f an topological product of LC spaces is LC; a morphism whose domain is LC is closed and in onto simply 2 or V . subcategory of B(K) by D(K) or simply . lVl, call 2 ( K) continuous a equipped -necessarily-with the disby K . A morphism V -K is called a The full ces we we mean dimensional space, one will also be denoted topology, crete of such spaces morphism so space V another space is an particular isomorphism. a 1-1 map from A discrete space is LC an LC space iff it is finite dimensional. The full subcategory of LC spaces is 4 denoted G( K ) , or simply S. The inclusion C -> B has that when K is a field, l->BV. V right adjoint, linear compactness is It is worth exactly the compactness. Thus the concept of linear compactness ological the extension as finite a field of notions arbitrary to an easily mentioning same as can top- be viewed described for finite ones. It is well known the that C and D of all functionals of V set product KV. Since K is LC are topologized (trivial), so V E 2, then V* E C is subset of the topological dual. If as is the a product. For any v2EV. vll is the inverse image of the discrete space and is hence closed. The intersection of these sets, for all v1 closed and so the functionals near is obtained set on of additive maps is closed. V is closed and is hence LC . The by sending the reverse set is of all li- equivalence to the discrete space of functionals topologize the space Hom(V, W) of contin- the LC space V V, W E B. We will Let linear maps in uous by letting where Wo as ways. We is an of open Wv . subspace in W . In fact, Since the latter is A basic open 11, subspace of [V, ",’ J Vo C V is an LC W] we are topologized separated, topologizing ( V , W ) so is ( V, W) . It is unless V is discrete. In closed where let ( V, W) denote that space subbasic subspace be a subspace a two subspace in general discrete and W is LC, (V, W) [ Similarly and v2 , it. on a two vectors subspace particular, is also LC . The second topology not if V is is finer. is and Wo C W is open. We call ( V, W) .,,and then weak and strong internal hom 5 respectively. The fact that is continuous follows from the fact that the ny We let V * denote PROPOSITION need that V / U 1.2. ( V , K ) and V " denote [V, K The space K is That it is now PROPOSITION first choose with dense not open the zero category 2 In the image regular are an a cogenerator quotient monos class. The existence of maps and are such the requir- theory. are monos inclusions are a cp E V* we C V with ve U. Then subspace 11 follows from standard linear 1.3. cogenerator in B. a if V c 2 and v # 0 in V , then there is is discrete and v+ U is ed functional generator and a generator is evident. To show it is a only show that O(v) + 0 . But ular epis ma- is LC . vectors PROOF. subspace spanned by finitely 1-1 maps,- of closed cpis arc suhspacr?s,’ maps reg- open,. the category i.s well- and cO-l1Jcll.. powered. PROOF. This is routine and is left PROPOSITION maps The category 2 is complete and cocomplepte. The are The topologized by the coequalizer of two is the cokernel of their difference. The cokernel of tient modulo the closure of the THEOREM. ses the reader. completeness is routine. Direct sums topology rendering the injections continuous. PROOF. finest 1.4. to image . projective and injective (withrespectto the classubspace iraclusions, respectivPl y ). T’he space K is of regul ar epis P ROO F . The and projectivity is trivial. Suppose V C W’ is a subspace q5: V -> K, As usual qb is uniformly continuous and that ql has a unique extension to the closure of V . Thus V to be closed. Let Vo ker O. Then Vo is closed in V pose K = We have that V/V0 ->W/V0 is continuous and V / V 0 is LC is an logy isomorphism from with the W/ Vo . Thus we that V = K v . Now there is zero on v map is the quo- a and a image, hence V / V 0 some suitable scalar functional defined multiple 6 of it is is discrete we one on O(v). so may suppose and hence in W . so still has the may suppose that V is and sup- that the map subspace dimensional, topo- in fact all of W which is non- 2. The weak duality. For 2. 1. P ROPOSITION a inverse limits. T’hus it has the the level of the At PROOF. is topology topologized gized a as issue. But at subspace a as of is of spaces, that is clear. vector is a (II Vi) V = II(VVi) topologized if and adjoint with the VV1, hence of of now Only (V, I1 Vi) is II (V, Vi) is topolo- V1 C V2 subspace as a with of spaces, family Similarly, commutes subspace V2 , follows from the hen- special functor theorem. T’he - -O- described above is PROPOSITION 2.2. Both maps U O V -> W and V O U -> W PROOF. U X V -> W. It is bilinear maps respond be the if {Vi} The existence of the (V, V2 adjoint underlying of the latter. subspace topology, ( V , V1) ce fixed V E B, the functor ( y’, -) an adjoint - 0 V - to the same set. only linear map. In order that corresponding necessary and sufficient that for each u In order that inverse Fl image {f be continuous on correspond to necessary Let F : U X V -> W be symmetric. a show that certain they set each of cor- bilinear map and F1 map U , F( u, -) E to a I ul to ( V, W) , it is be continuous on V . is necessary and sufficient that the U , it of lf( v) W0}, where v E c V, Wo is an open subspace of W, exactly the condition that for a fixed v , F(-, v ) Thus F corresponds to a map L’ ->( V, W) iff it is sepU and V which is evidently the condition to corres- be open in 11 . But this is be continuous arately pond to on a map V ->( U, W ) . COROLLARY 2.3. the on functional P ROO F. U . continuous in The natural map V* evaluation lVl -> V** l , is realized at v, In view of the symmetry above maps, the first being the identity : 7 we by which a have the assigns to v EV map V -> V **. following sequence of isomorphism we will say that V is weal. clearly that of the weak hom , simply reflexive. Yhen this natural map is reflexive or, if the context is THEOREM 2.4. PROOF. The an The natural map V -> V** is of hypothesis show it is onto, observe that always 1-1 and onto. separation is equivalent we have topologized V * to its as a being of KV. functional since K is is open, To subspace injective, any functional V* t-+ K extends to a f : KV -> K. Continuity together with discreteness of K implies that tors through the projection onto a finite number of factors. For Thus, 1-1 . and any open subset of the (necessarily open) set under the is the inverse product projection this fac- to a finite image of some product. So f has a factorization The second, being a functional finite dimensional space, is a on a linear combination of evaluations, say Applied att to a O E V*, that is, to a linear map, this is the same as evaluation a1 v1 +... +an vn. Hence whether or not reflexiveness reduces the purely topological question of open). example, an immediate conto V -> V** is closed ( or For sequence is: COROLLARY 2.5. If V is LC, COROLLARY 2.6. For any PROOF. which is right that is 1-1 and onto, they lVl -> onto reflexive. and 1-1. as always - are inverse A space V is called map is reflexive. V, V* is We have V -V** is inverse - V to Dualizing gives V** * -> V*, the natural map V* - V*** . Since isomorphisms. representationally K is continuous. us Evidently 8 a discrete ( RD ) if every linear discrete space is RD, but the con- Clearly the condition is equivalent to every cofinite dimensional subspace being open and that, in turn, implies that every subspace - being the intersection of the hyperplanes containing it-is closed. The minimal RD ( MRD ) topology is the one in which the open subspaces are precisely the cofinite dimensional ones. That clearly defines a topology, which, just as clearly, is discrete only in the finite dimensional case. fails. verse is MRD; an RD space is PROOF. iff its dual is LC; reflexive iff it is MRD. A space is RD THEOREM 2.7. the dual of an LC space RD, then V* is closed in K V (see the discussion follow- If V is above) and hence is LC. If V is LC, then from the known duality V*l*, whence the natural map V** -> I V*I * is 1-1 and onto. Thus V* ing 1-1 V 3’ ) l is RD. there is another RD the of points as V *. Call the resultant topology W . We can assume, by taking their in f (which can be immediately seen to give an RD topology) that the topology on W is coarser than that on V * , which means that V* - W is continuous. Both V** and W* are LC so that the map W* - V** , which is clearly 1-1 and onto (they are both the set of all functionals on IV *I - lWl), is an isomorphism. Suppose topology V*** -+ W** is also. The result Thus can now on be same set seen from the commutative square in which the bottom and left hand map is RD and V - V** continuous ently RD. RD, V -V** with reflexive, then V** = V If V is If V is RD and P ROPOSITION 2.8. PROO F. ce W means isomorphisms. If V* is LC, V** V has a finer topology which is evidV** MRD implies they are the same. If W C V and V is are is RD. reflexive, so is W. V * * - V , the points of W ** are carried into W. Sinsubspace topology, that defines a continuous map W** - W Under the map has the which is inverse to COROLLARY 2.9. the natural map. If V is reflexive, every RD 9 subspace o f V must be MRD . The space V is TH E O R E M 2. 10. finite subspace is that V is not dimensional. The PROOF. necessity be cannot of this condition is 2.9. Suppose now subspace of V which is not open in I’**. cofinite dimensional, for then it would be the intersection of reflexive and let W be W every discrete reflexive iff of the kernels of same functionals, crete spaces an open finite number of functionals. But V and V** have the a thus W is open in the latter. Now V/ W is discrete. Dis- are projective ensional discrete subspace. and so V - V/ W splits, giving V an infinite dim- 3. The weak hom . PROPOSITION in the of [2], sense a composition law (V,W) ->((U,V),(U,W)) l. 2. The function is the obvious PROOF. and with the pointwise would like to one. The only question is continuity hom that is easy. it follows that From this we There is 3.1. show that the full we have a subcategory closed monoidal category. We of reflexives is likewise. First need: PROPOSITION For any V and any 3.2. reflexive W, (V, W) is rcfl(,xi7,c. PROOF. and the result follows from 2.6. We now let BR(K) denote the full subcategory of rs ( K) consisting of the weak reflexive spaces. THEOREM 3.2. PROOF. while a The inclusion If W is reflexive, map V** -W TH EO R E M 3.3. When a gives, BR(K) CB(K) map V -> W up on has a le ft adjoint with equipped with ( - , - ) , BR (K) 10 l-> V**. gives composition category. V V -V**, a is a map V -> W. closed monoidal PROOF. e so reflexive spaces, are 0 V) ** is the adjoint that used If U , V , W in B (K) is of course to we have ( V , - ) . The same composition law as is intended here. 4. The strong dual. with the idea of better understanding vector discrete spaces. Although the theory developed in the prespaces, ceding two sections is attractive, it has the disadvantage of excluding the discrete spaces. It is true that the full subcategory of MRD spaces is equivalent to the category of discrete spaces, but nonetheless the theory leaves something to be desired. Additionally, experience tells us that pointwise convergence is the wrong topology for function spaces. Thus we are led to consider the strong dual and strong hom . Recall that [ V, W] is the space of homomorphisms which has, as a basis of open subspaces, We began including is where Vo V - the [ context LC an K]. V, this theory subspace in V and Wo is an A space V will be called strong is clear - provided the natural map open subspace of W . Also reflexive - just reflexive if l v l -> l v "l comes from an isomorphism V -> V ^^. As we will see in the next section, it will be necessary to restrict to a subcategory in order to make [-, -] into an internal hom. THEOREM The dual 4.1. o f an RD space is compact ; the dual ofa compúe space is discrete,. The second PROOF. know that But an an statement RD space cannot is obvious. The first is clear have an infinite dimensional LC space as soon as we infinite dimensional LC must have subspace. discontinuous functionals ; otherwise the associated discrete space would be self dual for the discrete dual, which is known to PROPOSITION The natural map open sets PROOF. 4.2. be impossible. i7i V "". Let U C L’ be open. Consider 11 I Vl -1 v--l takes open sets in V to Thus V/ U is discrete, so (V/U) ^ is is discrete. The LC, and then (V IV)"’" kernel of V ^^ ->(V/U)^^ is the image of U in V A: The kernel is the inverse image of the open set n in (V/U)^^ and is thus open. When U OF, Th(1 TH EO REM 4. 3. we natural map When V is PROOF. let U l C V^ denote the annihilator of I vil -j, V^l L.C, the functionals on is 1-1 and onto. V^ are the and hence the result follows from 2.4. In the V A --+K must, U’ the form O’: U" -+ K aluation in order for some LC c 11. Then at a u subspace is (continuity general not an so U C V . But a map. proof onto. V * on a O: of subspace a functional issue since U^ is discrete) which is ev- is 0. This is of course not a natural map V^->V the natural direction for such permits an easy of For dny V , Since any map from is those functional induces But it is the direction in which it is continuous and CO ROL L ARY 4.4. one case a then O The result of this and of 2.2 is that there is which is 1-1 and same as be continuous, contain in its kernel to U . an V^ is reflexive. LC space is closed and any map is open, it follows that discrete and LC spaces locally LC if it has open LC subspace. In discrete, hence projective, and thus the quotient is the direct reflexive. sum To see of a that an discrete and not an are every reflexive is discrete reflexive. A space that case map splits LC space. Such to a a locally LC, the quotient is and the space space is we clearly consider the following EX A M P L E a subspace 4. 5. Let V be the direct of the direct product W’ . sum of Ye have Clearly 12 Xo copies of K topologized as isomorphisms, V^ = and I claim that under these se V . For V is evidently den- in W , from which W ^ -> V ^ is 1-1. It is also onto, since every continuous homomorphism between topological groups is uniformly continuous and hence has a unique extension to the closure. V has no infinite dimensional LC subspace, since V has countable dimension and any infinite dimensional LC subspace, being the dual of a discrete space, must have dimension at least I Vl ^ V = 2 o . Thus V A is is also, it follows VA and V ^^ -> V is We say that THEOREM 4.6. open subspace PROOF. The isomorphism. an subspace on l V)l every functional clear that U is a topologized by pointwise convergence that V A has the subspace topology from l U C V is which vanishes representationally iff every RD equivalence not locally V I A. Thus LC. representationall y open provided on U is continuous on V . It is open iff V/ U is RD . The space V is strong is open Also V i s and since of the reflexive ill quotient two every representationally is discrete. latter statements is evident. If V is strong reflexive, so is every quotient (just turn around the proof of 2.8). But a reflexive RD must be discrete, since its dual is compact and its second dual discrete. Conversely, if V is there is U r V whose image a set V AA have the I vAAl same reflexive, V^^->V in V AA is open. Since functionals. Since U is open in which vanishes representationally not on open and U is continuous not open. 13 on V, VAA, hence is not open, V^= V AAA, V any functional on so and on V . Thus U is REFERENCES 1. J. DIEUDONNE, Introduction York, 1973. 2. S. EILENBERG Algebra the and G. M. KELLY, ( L a Jolla, 1965), 3. S. LEFSCHETZ; to pp. 421- Theory of formal groups, Closed categories, Proc. M. Dekker, New Conf. categorical 562 ; Springer, Berlin, 1966. Algebraic Topology, A. M. S. Colloquium Pub. XXVII, New York, 1942. ADDED IN PROOF. In a subsequent paper I will discuss the modifications necessary to make a Department of Mathematics Mc Gill University P. O. Box 6070, Station A MONTREAL, P. Q. H3C 3G1 CANADA 14 at length the strong hom and closed monoidal category using it.