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Transcript
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
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