Research Notes in Mathematics
MDeWilde
Closed graph theorems and webbed spaces
Pitman LONDON · SAN FRANCISCO · MELBOURNE
19
MDeWilde University of Liege
Closed graph theorems and webbed spaces
Pitman LONDON · SAN FRANCISCO · MELBOURNE
PITMAN PUBLISHING LIMITED 39 Parker Street, London WC2B 5PB FEARON-PITMAN PUBLISHERS INC. 6 Davis Drive, Belmont, California 94002, USA Associated Companies Copp Clark Ltd, Toronto Pitman Publishing New Zealand Ltd, Wellington Pitman Publishing Pty Ltd, Melbourne
First published 1978
AMS Subject Classifications: (main) 46A02, 46A30, 46F02 (subsidiary) 46A05, 46F05
© M De Wilde 1978 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,or transmitted in any form or by any means, electronic, mechanical, photocopying, recording and/or otherwise without the prior written permission of the publishers. The paperback edition of this book may not be lent, resold, hired out or otherwise disposed of by way of trade in any form ofbinding or cover other than that in which it is published, without the prior consent of the publishers. Reproduced and printed by photolithography in Great Britain at Biddies ofGuildford ISBN 0 273 08403 8
Preface
The first closed graph theorem was proved by Banach for
~-spa
ces, more than forty years ago. It became very quickly a central result in functional analysis and its applications. It has by now been generalized in many ways. The various generalizations essentially split in two main trends. On one side, the domain space is, for instance, assumed to be barrelled (or somethin~
connected to the Baire category property) and suitable
conditions on the range space are determined to insure the validity of the closed graph theorem. They turn out to be connected somehow to a completeness condition. A detailed account of these results can be found in [90]. Of course, for practical purposes, it is important to know whether the closed graph theorem is true for those locally convex spaces that occur in applications, such as the various spaces of distributions of L. Schwartz. The question was asked by Grothendieck in his thesis [45,p.19].
Observing that ultra-
bornological spaces would be a reasonably wide class of domain spaces, he conjectured the existence of a corresponding appropriate class of range spaces. The problem of finding an adequatly comprehensive class of range spaces was studied by Slowikowski [98,99], Raikov [87], L. Schwartz [94], Martineau [72,73,74] and the author [14,19,21]. The present notes are essentially devoted to the study of the author's webbed spaces and of the results of L. Schwartz and Martineau. The basic results on the domain and range spaces are developped in a self consistent way. A large part of the book is of graduate level. It also includes more specialized results, reaching the level of current research.
Chapter I begins with the closed graph theorem of Banach. Starting with the Baire category property, it develops the subject and its first easy extensions in a manner that traces the way towards further generalizations. Chapter II is concerned with the permanence properties of the domain and of the range spaces. The introduction of linear relations and the corresponding hereditary properties on the range space are due to Raikov. The reader who wants to restrict himself to the main results may omit this chapter, excepting prop.II.).2, which points out the interest of ultrabornological spaces as domain spaces. The theory of ultrabornological spaces is developped in Chapter III. Since it is entirely parallel to the study of bornological spaces, both cases are handled together. It is widely inspired by §28 of [62] and completed by results of Schwartz, Hogbe-Nlend and Grothendieck on ultrabornological dual spaces. Chapters IV and V present the theory of webbed spaces. We essentially follow [14], with various improvments due to the systematic use of fast convergent sequences. The basic results are in Chapter IV. Chapter V collects sophisticated permanence properties of spaces of linear maps, spaces of vector-valued functions and tensor products. Although the concept of webbed spaces has been extended to topological vector spaces, we restrict ourself to locally convex spaces. The reader will find in [90] an account of the theory for topological vector spaces. The fact that a linear map T is relatively open or weakly open, or that its range is closed, is closely related to similar properties of its adjoint T* and also to the possibility of lifting equicontinuous sets to equicontinuous sets by T*. In suitable cases, this lifting problem can be solved by means of webs. Chapter VI outlines these questions, which play an important role in applications as shown in [104,105].
The last chapter is devoted to the Borel graph theorem of L. Schwartz. After recalling the required properties of Souslin spaces, we present the proof of Martineau and another one due to Christensen. The results are then extended to topological groups and various permanence properties for spaces of linear maps are sketched. I want to conclude by thanking J. Schmets for his helpful remarks and careful reading of the manuscript. I am also indebted to Mrs Streel for typing the first draft and the camera ready manuscript. Marc De Wilde
Terminology and notations
The main part of these notes is concerned with Hausdorff locally convex topological vector spaces (l.c.s). Our notations are pretty standard. We call an U-space a metric complete l.c.s. A
L~-space
is a countable inductive limit of such spa-
ces. A countable inductive limit of l.c.s. E. (i E-lN) is strict ~
if the topology induced by Ei+ 1 in Ei coincides with the topology of E ~.• Let E be a l•c.s. If A C E, we denote by Co A, Co A and >A<, its absolutely convex, closed absolutely convex and linear hulls. The~
of a l.c.s. E is denoted by E 1
,
its algebraic dual
by E*. The polar of A C E in E 1 or of A C E' in E is denoted by A We denote by a(E*,E) the topology (on E*) of pointwise con0
•
vergence on E. A collection
=
~
of bounded subsets of E is called a covering
and if, given B1 , B2 E- ~. there exists C ) 0 and B3 E- ~ such that B1 U B2 C C B3 • If ~ is a covering family, we define on E 1 the topology t~ of uniform confamily if E
UBE-~B
c;
vergence on the elements of topology
t~
~.
, it is denoted by
covering families
~
When E' is equipped with the E~
• The main examples of such
are
~b
the set of all finite subsets of E, the set of all bounded subsets of E,
~pc
the set of all precompact subsets of E,
~c
the set of all compact subsets of E,
~ca
the set of all absolutely con vex compact subsets of E, the set of all absolutely convex subsets of E*, compact
~s
~
't
for a(E,E 1 ) .
When it is useful, we write ~(E) to mention the space E. The space
E', equipped with the topology
the above
~·s,
E sI
'
Thus E~
corresponding to
is denoted
Eb' ' E'pc
=
t~
'
E'
c
'
E' ca '
E'
-r
E~(E',E)"
The topology -r is called the Mackey topology of E and E is a Mackey space if it is equipped with its Mackey topology. Let E,F be l•c.s. A linear map T of E into F is generally defined on the whole of E.
Otherwise it is explicitly mention-
ned and the domain of T is denoted by ~(T). Its range {Tx:x f
E}
is denoted by R(T) and its graph by G(T).
Contents
I. CLOSED GRAPH AND OPEN MAPPING THEOREMS FOR
~-SPACES
1. Meagre sets and Baire 1 s category theorem
1
2. The closed graph theorem for
4
3. The open mapping theorem
~-spaces
.............
10
II. HEREDITARY PROPERTIES FOR CLOSED GRAPH THEOREMS 1•
Introduction
2.
Linear relations
3. The domain space 4. The range space
..... . . . .. . . ............. ...... .............. ... ...
....... ......
12 12 14 24
III. BORNOLOGICAL AND ULTRABORNOLOGICAL SPACES 1•
Fast convergent sequences
29
2.
Bornological and ultrabornological spaces
34 39
3. Permanence properties and examples IV. WEBBED SPACES
...... . ... . .. .. .... ... .........
1. Webs 2. A topological interpretation
.......
..
48 52 56 58
3. Some examples •••••••• 4• Permanences properties ••• 5. Closed graph theorems •••••••
63
6. Localization properties
69
V. WEBS IN SPACES OF LINEAR MAPS ••••••••••••••••••••••••••
76
2. Tensor products •••••••••••••••••••••••••••••••• ,3. Spaces of vector valued functions ••••••••••••••
89
4. Localization properties for subsets of L(E,F)
99
1. Spaces of linear maps
92
VI. HOMOMORPHISMS AND WEAK HOMOMORPHISMS 1. Homomorphisms and duality
0
•••••••••••••••••••••
10.3
2. Surjectivity theorems •••••••••••••••••••••••••• 110 .3. Lifting compact subsets •••••••••••••••••••••••• 11.3 VII. SOUSLIN SPACES AND THE BOREL GRAPH THEOREM OF L. SCHWARTZ 1. Souslin spaces
•••...........••................•
2. Permanence properties and examples ,3. Some properties of meagre sets
126
••••••••••••• 128
••••••••••••••••• 1.3.3
4. The Borel graph theorem for l.c.s. ••••••••••••• 1.36 5. The Borel graph theorem for topological groups 14.3 6. Further permanence properties of Souslin spaces BIBLIOGRAPHY INDEX
...........................................
145 149 1 57
I Closed graph and open mapping theorems for 7-spaces 1. MEAGRE SETS AND BAIRE 1 S CATEGORY THEOREM DEFINITIONS I.1.1. Let X be a topological space. A subset e of is 0
e
=
¢.
~
if its closure in X has a void interior, i.e. if
A subset e of X is meagre if it is the union of counta-
bly many rare subsets of X. A topological space X is a Baire space if no non empty open subset of X is meagre. It is obvious that a subset of a rare or of a meagre subset of X is itself rare or meagre. Thus, in the definition of a meagre subset, it is equivalent to assume that it is contained in a countable union of rare subsets. PROPOSITION I.1.2. A countable union of meagre subsets of X is meagre. Straightforward. PROPOSITION I.1.3.
1! e is non meagre, the interior of its
0
closure,
e,
is non empty.
Indeed, e is not rare. PROPOSITION I.1.4.
1! X+¢ is a Baire space and if it is the
union of countable many closed subsets, then one of them at least has an interior point. Indeed,
otherwise X would be meagre.
REMARK. We adopt here the Bourbaki terminology. The original terminology, still used by many authors, is nowhere dense
(= rare), of the first category (= meagre) and of the second category (= non meagre). This makes clear the name of the
following theorem. BAIRE 1 S CATEGORY THEOREM I.1.5. A complete metrizable topological space is a Baire space. Although the result is well known, we mention an interesting proof, due to G. Choquet [11,12]. PROPOSITION I.1.6. ~X be a topological space. It is a Baire space if there exists a binary relation between the non empty open subsets of X, denoted [, such that (a) (b) (c) (d)
w c w' [w" c w'" w [w' => w c w' , Vw, 3w 1 : w' [w wn+1 [wn' vn E- lN
~
w [ w'" '
, =>
n nE-lN
wn 1=¢ •
Indeed, assume that a non empty open subset w is meagre, hence contained in the union of ~ountably many closed subsets Fn(nE-lN) with void interior. Since F1 =¢,we have w j F 1 • There exists thus an open subset w1 .f=¢ of w such that w1 n F 1 = ¢. By (c), there exists w1 such that w1 [w 1 thus w1 [w by (a). Again, w1 j F 1 U F 2 , thus there exists w2 such that w2 n (F 1 uF 2 ) = ¢ and w2 [w1• By induction, there exists then a sequence w'n (n E-lN) such that w'n+ 1 [w'n and
w'n n [F 1 u ••• uFn] =¢for each n E-IN. By (d) 00
¢
I= n
n=1
w'n
c
w
•
However, 00
00
( n WI) n ( u Fn) n n=1 n=1
¢
,
hence a contradiction. Proof of prop. I.1.5. If X is complete and metrizable, let d be a distance compatible with the topology of X; define diam e = inf {1, 2
sup d(x,y)} x,yE-e
•
Then the relation w [w' ~> ~
< w'
and diam w ~
I
diam w'
verifies the above conditions. For {c), observe that
<
{y : d(x,y)
£/2)
[
{y : d(x,y)
<
£), Vx, V£
<
1/2.
-n+1 For ( d ) , if wn+ 1 [wn for each n, then diam wn ~ 2 • Take then x n E- wn for each n E- lN. The sequence x n (n E-lN) is clearly a Cauchy sequence. Since X is complete, it converges to some
Since xn E- wn
Xo
for n 0
00
X
n o' we have
X
E- wn
. Thus
0
00
n
E-
~
n=1
wn
(
n n=1
wn •
Hence the conclusion. REMARK. The interest of this proof lies in the fact that it applies to
~any
other situations. To quote another simple
example, let us mention that a locally compact space is a Baire space. This is straightforward by equipping X with the relation
w [w'
~> w
< w'
and w relatively compact.
DEFINITION I.1.7. Let E be a l.c.s. A barrel in E is a clo00
u ne. The space n=1 E is barrelled if every barrel of E is a neighborhood of zero. sed absolutely convex subset e such that E =
COROLLARY I.1.8. Every U-space is barrelled. Indeed, let E be an U-space and 9 a barrel in E. We have 00
u ne. Moreover E is a Baire space and ne is closed for n=1 0 each n. Thus ne ~ ¢ for some n. Since e is absolutely convex,
E 0
9 is also absolutely convex, thus it contains O, hence the conclusion. Some deeper properties of meagre sets will be developped in chap.VII, §3.
3
2. THE CLOSED GRAPH THEOREM FOR PROPOSITION I.2.1. E
~
~
E, F
~-SPACES.
~~-spaces
and T a linear map of
F. If the graph of T is closed in E x F, then T
~
continuous. Denote by Ui (i ~:m) and Vi (i ~lN) a countable base of neighborhoods of 0 in E and in F. We may assume that ui+1 + ui+1 c ui and vi+1 + vi+1 each vi is closed.
c
vi for each i ~ lN and that
Step a. For each k we have 00
E
= T -1 F = u
n=1
n
T- 1 V
k
00
C
u n=1
n
T
-1
(1)
Vk • -1
Since E is a Baire space, by prop.I.1.4, some n T Vk has an interior point hence, by the argument of cor.I.1.8, it is a neighborhood of zero : there exists ik ~ :m such that
Comment. We can read (1) as follows : take ~bsolutely convex neighborhood of zero in F, say V. Then T- 1 v is a barrel in E. Thus the assumption "E barrelled" would suffice for ste'P a. Instead of doing so, we have implicitly reproduced the proof that a Baire space is barrelled. But then, by prop.I.1.2, we have some more information on T- 1 vk : it is non meagre. This is of no use in this context but will be essential later on. Step b. Assume that, using ste'P a, we have determined an increasing sequence ik such that
(2) The point now is to get rid of the closure in the right-hand side. Indeed, without closure, (2) would mean that T is con-
4
tinuous. It follows from (2) that U.
~k
C T
Take xk
-1
V k + U.
~k+1
1fk •
• There exist yk
E- U.
~k
0
,
0 0
such that
So we can find inductively a sequence xk (k>k 0
)
and yk(k~k 0 )
such that E-
u.
k
~k
0
Summing up both sides of xk we get
thus
On the other hand, n n T( I: yk) = I: k=k 0 k=k
Tyk 0
Since Tyk E- vk for each k, the series
Tyk is a Cauchy
I: k~k
0
sequence hence, since F is complete, it converges to some point z. Moreover, n I:
k=k 0
Tyk E-
n I:
k=k 0
'In
'
thus z E- vk -1" 0
Now, since the graph of T is closed in E x F,
5
n I:
k=k 0
Yk -
xk 0
n
T ( I:
k=k
0
was an arbitrary element of U.
Since xk
, we have thus
l.k
0
0
proved that
hence the conclusion. Comment. The metrizability of E and F was strongly used through the existence of countable bases of neighborhoods of zero in both spaces. The completion of F was used for the convergence of I: Tyk' k
It is however not difficult to improve prop.I.2.1 in two ways. PROPOSITION I.2.2. Prop.I.2.1 holds true when E is a barrelled space. Take as above a countable base V. (i E-lN) of neighborhoods of l.
zero in F. We have already observed that step a holds true when E is barrelled. It provides a sequence Ui (i E-lN) of neighborhoods of zero in E such that U.l. C T
-1
V.l. , Vi E-lN •
It is of course not a restriction to assume that Ui+ 1 + Ui+ 1 CUi for each i f lN. However, the U. 1 s no longer form a base of l.
neighborhoods of zero in E. As in step b, starting from xk
f 0
Uk , we determine 0
xk+ 1 ,yk (k~k 0 ) such that xk = yk + xk+ 1 , xk f 6
Uk' TykE- Vk, Vk
~ k0
•
The series
Tyk still converges to z
I: k~k
E-
0
vk -1 and, moreover, 0
n I:
xk
k=k 0
0
yk +
X
(1)
n+ 1' Vn •
Let U be an arbitrary neighborhood of zero in E and fix n f IN • By ( 1 ) , n
+ T
E yk k=k 0 + T
-1
Vn+1 +
-1
V
n+ 1
u •
Hence there exists z 1 such that n
E yk - z 1 k=k 0 and Tz' n
E-
E
k=k 0
E-
x
ko
( 2)
+ U
v n+ 1 • On the other hand, Ty,_ 1\.
-
00
z =
E
k=n+1
Tyk
E-
v n+1 + v n+2 + •••
c
v n"
Thus
( 3) Combining (2) anc (3), we see that (xk ,z) is in the closure 0 of G(T) in E x F, hence Txk = z, which yields thE conclusion. 0
There is another interesting proof of prop.I.2.2, based on duality arguments. DEFINITION I.2.3. If T is a linear map of E into F, the transpose T* of T is the linear map of F* into E* (algebraic duals of F and E respectively) defined by <x,T*y'>
=
E-
E, Vy'
E-
F* •
Denote by ~(T*) the subset of all y' E- F' such that T*y' E- E•. It is the domain to which T* must be restricted to define a map of F' into E•.
7
~
PROPOSITION I.2.4o into F.
JL
E,F be l.c.s. and T a linear map of E
E i s a Mackey space and if ~(T*)
=
F', ~ T ~
continuous. X
Indeed, for every The map T* is continuous of F's into E'. s E- E, <x,T*y'>
1 )
is a continuous form on F's" Let now V be an arbitrary absolutely convex and closed
vo
is compact in F' s ' is compact and absolutely convex in E~ and
neighborhood of zero in F. Its polar T*V 0 T- 1 v
=
(T*V 0
) 0
so
is a neighborhood of zero in E. ~
PROPOSITION I.2.5.
T be a linear map of E
~
F.
Then the graph ofT is closed if and only if ~(T*) ~ o(F 1 ,F)dense in F 1 • The polar of G(T)
{(x• ,y') E- E' x F'
<x,x'> +
O,
'tfx
E-E}
is clearly equal to {(-T*y 1 ,y
1 ):
y' E- ~(T*)}
Thus (x,y) E- G(T)
00
~
)
~
) Tx-y E- ~(T*)
~
>
(Tx,y
1 )
-
O,
1 )
'tfy
E- ~(T*)
0
and G(T)
00
= G(T)
~(T*)
0
= {o}
hence the result. We shall need the Krein-Smulian and Banach-Dieudonne theorems [51, p.246; 43, p.234]• PROPOSITION I.2.6.
~
E be an
~-space.
A convex subset e
~
E' is closed for the topology o(E' ,E) if and only if, ~ every neighborhood U of zero in E, e
n
is closed for o(E 1 ,E). If E is metrizable, a subset e of E' is closed in E' i£ pc 8
U0
and only if, for every neighborhood U of zero in E, e
n uo
is closed for a(E•,E). Second proof of prop.I.2.3. By prop. I.2.4, it is enough to
= F 1 • The assumption that G(T) is closed
prove that ~(T*)
means, by prop. I.2.5, that ~(T*) is dense in F 1 for a(F 1 ,F). Thus we are left to prove that ~(T*) is closed in F• for 0
(E•,F). By the Krein-Smulian theorem, this is true if
~(T*)
n
yo
is closed for a(E ,F) for every neighborhood Y of zero in F. 1
We have
Indeed, the left-hand set is closed in E and moreover it contains T
-1
X f
Y : T- 1 Y, y' f ~(T*) n yo ~ l<x,T*y'>l
-
Since E is barrelled, T T*[~(T*)nY
Thus in E
1
0
]
-1
= I
l ~
1.
Y is a neighborhood of zero in E.
is equicontinuous, hence relatively compact
for a(E ,E). Now, T* is obviously continuous ofF* 1
equipped with the topology a(F*,F) into E* equipped with the topology a(E*,E). So it maps ~(T*)
n yo
a(F*,F) (1)
into E 1 • This means that (1) C ~(T*), hence the conclusion. Comment. The proof has been written in such a way that it points out further steps toward generalization. It is clear that F does not need to be an
~-space
but only has to verify
the Krein-Smulian property. Even this is too much : the subset ~(T*) for which this property is used is more than convex
1
it is a vector subspace. Another remark is that the barrelledness of E is only used to prove that T- 1 Y is a neighborhood of zero. So it could be
9
avoided by strengthening the assumptions on T. The generalization of the closed graph theorem along these lines was inaugurated by Ptak
[84, 85].
This work raised up a
rather wide literature, which is out of the scope of the present notes. The papers of Ptak
[84, 85],
Adasch [1, 2, 3]
Valdivia [106, 107] and W. Robertson [90] will bring the interested reader to some of the nicest results and to a detailed bibliography. See also
[54].
More recent developments
appear in [31, 32, 33, 34] and [92, 93]. Despite its beauty, Ptak's theory and its further developments have not been of a great help to decide whether the closed graph theorem was true for the classical spaces of functions and distributions encountered in the applications of functional analysis. Another interesting generalization of prop. I.2.1 (i.e. the closed graph theorem for
~-spaces)
was given by Dieudonne and
Schwartz in [30], where they proved it for
~-spaces.
3. THE OPEN MAPPING THEOREM DEFINITION I.3.1. Let E,F be l.c.s. spaces and T be a linear map of E into F. The map T is open if Tw is open in F for each open set w C E. If T is bijective, T is open if and only if T- 1 is continuous. An open mapping T is always onto. Indeed, TE is open in F and it is a vector subspace. So it is equal to F. The map T is almost open if, for every neighborhood U of zero in E, the closure TU of TU induces a neighborhood of zero in TE. If T is continuous and open, it is called an homomorphism (or also a strict morphism). PROPOSITION I.3.2. (Banach's homomorphism theorem or open mapping theorem). Let E,F ~~-spaces. A continuous surjective map of E
10
~
F is an homomorphism.
The kernel ker T of T is closed since the graph of T is closed in E x F and ker T x [o} = G(T)
n
{Ex[o}).
Thus E/ker T is still an ~-space
-
If T is the map cano-
nically induced by T on E/ker T, T is still continuous, hence G(T) is closed. Moreover, T is injective. Its inverse T- 1 has a closed graph. Hence, by the closed graph theorem, it is continuous, which means that T is open. Comment. We have deduced the open mapping theorem from the closed graph theorem. The converse can also be done : assume that the linear map T of E into F has a closed graph G(T) and that E and F are
~-spaces.
Then, as a subspace of
E x F, G(T) is again an ~-space. The projection (x,Tx) -
x
of G(T) into E is surjective and continuous. So it is open {open mapping theorem). Its inverse x - (x,Tx) is thus continuous and also T. Prop. I. 3. 2, is more over true if, instead of being continuous, T has a closed graph. Indeed, in its proof, we only used the facts that ker T is closed (a consequence of the closedness of G(T)) and that G(T) is closed in E/ker T x F. Clearly (E/ker T) x F
=
{ExF)/(ker Tx[o}). Since ker T x {0} C G(T),
G(T) = G(T) + ker T x {ExF)/(ker T x
[0}).
{0} and so it has a closed image in But its image is G(T).
This way of connecting both results fails however in more general situations : it rests on the fact that the assumption on E and F is the same and is inherited by the product space E x F. So it wculd generally go wrong if E and F were not of the same type. Another proof is easily obtained by adapting the proof of prop. I.2.1 : simply interchange E and F and replace backwards images by direct images. Both proofs can be unified by considering "multi-valued mappings" or linear binary relations. This will be developped in chap. II, §2. 11
II Hereditary properties for closed graph theorems 1. INTRODUCTION Consider the class of all spaces E such that every linear map of E into a given space F is continuous whenever its graph is closed. Under which operations is that class closed? Of course, we can change the question by considering the open mapping theorem, by considering a class of spaces F instead of a single one, or by interchanging E and F. To set the problem straight off in its most general formulation, let us first introduce the concept of a linear relation. 2. LINEAR RELATIONS DEFINITIONS II.2.1. Let E, F be l.c.s. A linear relation of E into F is a binary relation R, which graph G( R )
co
{ (
x , y ) E- E x F : xRy }
is a vector subspace of E x F. We set R(x) R(e)
{yE-F : xRy}, (xE-E)
=
U R(x) , (eCE) • xE-e
. . Th e 1nverse re 1 a t.1on R- 1 1s aefined from F into E by xRy ~ ) y R- 1 x (xE-E, yE-F). Its graph is the graph of R, up to the order of the factors. Moreover, (R- 1 )- 1 = R. The linearity of R means that
12
The domain of R is
~(R) =
R- 1 (y)
U yH
Its range is R(E). If R is a linear relation of E into F and R' a linear relation ofF into G,,. t.h.§l product RoR 1 x RoR'z
~>
3y
,
defined by
F : xRy and yR 1 z ,
~
is obviously a linear relation of E into G. If T is a linear map of a vector subspace ~(T) of E into F, the associated relation RT defined by
is linear. Its graph is the graph of T. Moreover, 1
R; ( y) = T
-1
y, 'tJy ~ F •
The linear relation R is continuous if, for every open set w C F, R -1(w) is open in~ ( R ) • It is open if, for every open set w C E, R(w) is open in F. PROPOSITION II.2.2.
~
T be a linear map of E in1s F. It is
continuous if and only if RT is continuous. It is obvious, since R; 1 (w) PROPOSITION II.2.}.
_,
~
open if and only if RT Indeed, Tw
= T- 1 w for every w.
T be a linear map of E
~
F. It is
is continuous.
= RT ( w ) = ( RT-1 ) -1 ( w) •
PROPOSITION II.2.4.
~
E, F
~
V-spaces and R a linear rela-
tion of E 1n1s F with a closed graph. If its domain ~(R) 1A equal to E, it is continuous. If its range R(E) is equal to F, it is open. By means of prop. II.2.2 and II.2.3
1
this unifies the clo-
sed graph and open mapping theorems for V-spaces.
Since the domain of R is the range of R
-1
and conversely,
the two assertions are equivalent. The proof of either one is obtained by reproducing the proof of prop. I.2.1, replacing "Tx T- 1 e by R(e) or R- 1 (e).
= y" by xRy and Te or
3. THE DOMAIN SPACE DEFINITION II.3.1. Let us denote by
~F
the class of all spa-
ces E such that every linear relation R of E into F such that ~(R)
= E and G(R) is closed in E x F, is continuous.
PROPOSITION II.3.2. Every inductive limit of elements of
~F
belongs to ~F. So ~F is stable under inductive limits. It is an obvious consequence of the next two simple observations. Let E be the inductive limit of a family of l.c.s. E a(aH) and let R be a linear relation of E into F. If the graph of R is closed in E X F, the graph of its restriction G(R ) a
R to each F is closed in E a a a
X
F. Indeed,
= G(R) n (E a xF)
is closed in E
a
X F when E
is endowed with the topology in-
a
duced by E and this topology is coarser than the initial topology of E • If R
a
is continuous of E
a
into F for each a f
I, then R is
continuous of E into F. Indeed, if w is an absolutely convex neighborhood of zero in F, R- 1 (w) is absolutely convex and it induces a neighborhood
o~
zero in each E • Thus it is a neigha
borhood of zero in E. Comment. An obvious corollary of prop. II.3.2 is the following. Let ~ be a class of l.c.s. such that the closed graph theorem holds true for every linear map of a Banach space into a mem-
14
ber of
v.
Then it is also true for every linear
rna~
of an ar-
bitrary inductive limit of Banach spaces into a member of
v.
DEFINITION II.3.3. A l.c.s. E is called ultrabornological if it is the inductive limit of a family of Banach spaces. An ultrabornological space is obviously barrelled and bornological but the converse does not hold. However, the gap between ultrabornological and barrelled spaces is rather small. We have seen in Chapter I that barrelled spaces seemed to be a good generalization of V-spaces as domain spaces in a closed graph theorem. Since most usual barrelled spaces are in fact ultrabornological, it is actually worth restricting the generality on the domain space by assuming that it is ultrabornological if this restriction results in significant progress on the range space. By prop. II.3.2, it is equivalent to restrict the domain space to the class of Banach spaces. Rather neglected for years, ultrabornological spaces became very popular in the last decade. The reader who wants to restrict himself to the key results may skip over the end of this chapter, make his mind about ultrabornological spaces in chapter III and move to chapter IV. The next permanence property concerns products of l.c.s. PROPOSITION II.3.4. The product of a members of ~F
~F
belongs to
belongs to ~F. Before proving prop. II.3.4, let us first develop some general considerations about product spaces which provide an easy
i£ma
approach to the study of many permanence properties of such spaces. The material below is taken from [23]. DEFINITION II.3.5. Let E be the product of the l.c.s. E.(i~U). 1 For each x in E, xi is the component of x in Ei. If v C u, define x by (x ). = x. if i ~ v, 0 otherwise. We say that x v
v
1
vanishes on v if x
1
v
= 0. We call factor subspace
of E each
15
vee toT' subspace Ci = {x f E : xfl \ {i} = 0}, equipped with the topology induced by E. Each Ci is isomorphic to the corresponding Ei. For each x f
E, let us denote by E X the product of the
linear hulls of the x{i}(iE-0). If xi =f= 0 for each i f fJ, we say that E is a simple subspace of E. It is obvious that X
every simple subspace, equipped with the topology induced by E, is is om orphic to lRfJ (or CCV). LEMMA II • 3 • 6 • Given a seq u en c e x ( m) ( m E-lN ) .2£ E =
n iff/
E.l . such -
(m) is not equal to at most one of the xi 0, there exists a simple subspace R containing the sequence X x(m)(m E-JN). (m) (m) x. Take xi i f xi =f=o for some m and xi f Ei \ {o} l. otherwise. that, for each i
o,
f
PROPOSITION II.3o7o
~
U be a family of subsets of E =
n
Ei
HO such that, for each U f U, there exists V f U such that
v
+
v c u. l!t for each simple subspace Ex .2£ E and for each U E- U,
there exists a finite set v C 0 such that y f
E
X
t
Y
V
a
0 -) y
f
U '
then, for each U E- U, there exists a f.inite set v C fJ such that
y
v
=
0
~
y f
u •
Assume that the assertion is false for U set v C o, yv
=
0
~ y E- U •
Take V E- U such that V + V C
16
u.
1
for every finite
We construct by induction a sequence of finite subsets v (m ~lN) of fJ and a sequence x(n)(n ~lN) with the following m properties vm n v n = (J, (m) (m)
( i) (ii)
X
(iii)
X
XV
(m)
v,
(:
m
'
\tm
=/=n
\tm
'
\tm
.
Assume that vk (k<m) are defined. There exists x (: U m-1
U
vanishing on
vk. Let E
k=1
There is a finite subset z
~
O, z
~
(
)
Define then x m
be a simple subspace containing x.
y
E
x
=
z
~
y
~
~
of fJ such that
~
V •
and v
m
=
~\
m-1 U
k= 1
we have x(m) (: V, otherwise x would belong to x(m) and vm n v = (J if k < m. vm k To construct x( 1 ) and v 1 , start with v 0
x(m)
(
vk. Since x
u.
=
)
x m +xfl\ , ~
Moreover,
=
= (J.
Now, since the vm's are disjoint, there exists a simple subspace E containing the sequence x(m) (m ~lN) (by lemma X
II.3.6). There is a finite v such that y
V
= O, y
~
E
X
=9 y
~
V.
Since the vm are disjoint and v is finite, for m large enough, we have v n v = (J, thus x(m) = 0 and x(m) ~ V, hence
m
v
a contradiction. COROLLARY
II.}.s • .!&! t be a vector topology on E
=
n
EJ..•
i~fJ
If the topology induced by t
on the factor subspaces and the
simple subspaces is coarser than the one induced by the product topology of E, 1 ogy of E.
~
t is coarser than the product topo-
A base U of balanced neighborhoods of zero for t clearly satisfies the assumptions of prop. II.3.7. Let U ~ U be given.
Choose V
u
~
such that V + V C U. There is a finite set v C V
such that X
l>x~V.
0
v
n, take now V 1 ~ U such that V' + ••• + V•
If card v
(n times)
( v. eac~
For
i
~
v, V' contains a neighborhood wi of zero in
&i. Thus
n
Ei
ifv
X
n
wi (
v
+
v•
+ ••• + V 1 (n times)
(
u •
iE-v
REMARK. Corollary II.3.8 is true even if the Ei's are topological vector spaces, rather than locally convex ones. COROLLARY II.3.9.
~
R be a linear relation of E =
n E.1
HV
into a l.c.s. F such that ~(R) = E. If the restrictions of R to the simple subspaces and tha factor subspaces of E are continuous,
~
R is continuous of E illiQ F.
Let V be an absolutely convex neighborhood of zero in F. Denote by t
the locally convex topology on E for which 1 R- (V)
{2-m R- 1 (V) : m ~ m} is a base of neighborhoods of zero. By the assumption on R, the topology induced on the factor and simple subspaces of E by t
1
R- (v)
is coarser than the topolo-
gy induced by the product topology of E. Thus, by cor.II.3.8, t
R - 1 (V)
is coarser than the product topology of E. This yields
the continuity of R. Proof of prop. II.3.4. l-et
Ei(i~a)
belong to
~F
linear relation of E = n.L E.1 into F such that 1.,-a G(R) is closed in E x F.
and R be a ~(R)
=
E and
The restriction of R to the factor subspaces or the simple subspaces of E are again linear relations, defined on the whole of the space and having a closed graph. Since the factor subspaces and the simple subspaces, isomorphic toma, belong
18
to
~F'
the restrictions of R are continuous. The conclusion
follows by cor. II.3.9· PROPOSITION II.3.10. ~
~
E be a l.c.s. and La vector subspace
E of finite codimension.
~'
iL E belongs to
~F'
L belongs
,!s ~F" A trivial induction process reduces the proof to the case where L is of codimension one. Let R be a linear relation of L into F such that ~(R)
= L
G"[R) = G1 of G(R) in of a new linear relation R 1 of E into F.
and G(R) is closed in L x F. The closure E x F is the
gra~h
Since G1
n
(LxF) = G(R) ,
R' is an extension of R (in the obvious sense that xRy
=>
xR 1 y ) • Either ~(R 1 ) = ~(R) = Lor ~(R 1 ) =E. In the first case, G(R) is closed in E x F. If x 0
f
~(R),
there is a unique linear relation R" extending R and such that R"(x 0
)
= o.
Its graph is closed in E x F, since G(R) is
of codimension one in G(R"). In both cases, we have extended R to E by a linear relation with domain E and closed graph. By assumption, it is continuous of E into F, hence its restriction R is continuous of L into F. DEFINITION II.3.11. Anticipating somewhat chapter III, let us define a fast sequentially closed subset of a l.c.s. E as a subset e which contains the limits of all fast convergent sequences of E contained in e (see def. III.1.3). Let then ~F be the class of all spaces E such that every linear relation of E into F, such that ~(R) s E and G(R) is fast sequentially closed in E x F, is continuous. Define also &F (resp. &F) as tl1e class of all l•c.s. E such that every linear map of E into F is continuous if its graph is closed (resp. fast sequentially closed). 19
II.~.12.
PROPOSITION
f>F'
~
8F
8
Prop. II.3.2 and II.3.4 remain true for
F·
The proofs are entirely similar. However, the analogue of prop.
II.~.10
for
f>F
is unknown.
It is essentially because the arguments using the closure no longer hold. There is a partial converse to prop.
II.~.10,
due to
Iyahen [60]. However, we quote it here under the additional assumption that F is complete which seems necessary in its proof. PROPOSITION II.~.1~ [60). ~ E be a l.c.s., La subsuace of E of finite codimension and F a complete l.c.s. 1L L belongs ~
8F' ~ E belongs to &F. We may assume that L is of codimension one. Let T be a linear map of E into F such that G(T) is closed
in Ex F. Its restriction TIL to L has a closed graph in L x F, so it is continuous. Assume that L is dense in E. If x 0 f
E, the filter of
neighborhoods of x 0 in E induces a Cauchy filter
~
in L. By
the continuity of TIL' TIL~ is thus a Cauchy filter in F. Thus it converges to some y 0 f F. Since G(T) is closed, (x 0 ,y 0
f
)
( x 0 , Tx 0
)
..,..,..,..,......,..,ExF C G(T) and Tx = y 0 • Thus G(TIL) 0 ExF ExF -::-Gr.(T:"1~-L~) f ~Gr.(T:"11r-L"T) and G(T)
On the other hand, TIL has a unique continuous extension ~~~ExF
T 1 toE, with graph G(TIL) • Thus T = T• and Tis continuous. If L is closed in E, E is isomorphic to L x JR (or
~
• 12 , E f
8F for every F.
Thus, by prop.
8F •
Converse to the above results, the next properties reduce the size of f>F when F ranges over sufficiently many spaces.
20
DEFINITION II.3.14. If U is a collection of l.c.s., define
It is clear that
l£ U is the class of all finite dimensioOU are equal to the class of all l.c.s.
PROPOSITION II.3.15. nal l.c.s.,
~U ~
*For the same
u,
~~
and
OU
are the class of spaces E
~
that E£c is complete. Let E,F be l.c.s., the dimension ofF being finite, and T a linear map of E into F such that G(T) is closed. Then T is continuous. Indeed, let U be a base of neighborhoods of zero in E and p a norm in F. If T is not continuous, {y E- TU : p(y)
>
1}
+ p,
1/U E- U
•
Thus X
fp(Tx)
x
e- u,
p(Tx)
> 1Jue-u
is a basis of a filter converging to zero in E. Its image by T is a basis of a filter contained in the compact set {y: p(y) = 1}. Soithasan adherent pointy. Then (O,y) E- 'G"('T), but y
=/= o, thus ( O,y) f G(T). Hence the first assertion.
*For the second one, we may assume without restriction that F = ~. A linear map of E into F is thus a linear form. If its graph is fast sequentially closed, for every Banach disk B of E, the graph of TIEB is closed in EB~ (since EB and F are Banach spaces, EB x F is a Banach space). Thus TIEB is continuous of EB into F. Therefore T is a fast bounded linear form on E (cf. def.III.2.6). So, in the space E, every
21
fast bounded linear form must be continuous. By prop.III.2.7, it is equivalent to say that E}c is complete. PROPOSITION II.3.16. l !
&u
~ &~
u
is the class of all
l.c.s.,~u,
~u,
are the class of all l.c.s. equipped with their fi-
nest locally convex topology. Let E belong to &U and let t be the finest locally convex topology on E. The identity map of E into Et has a closed graph, thus it is continuous, and E = Et. Conversely, equip E with its finest locally convex topology t. If R is a linear relation of E into a l.c.s. F such that ~(R)
= E, let tR be the coarsest locally convex topology on E
which makes R continuous : if U is a base of neighborhoods of zero in F, {R- 1 (U) U E- u} is a base of neighborhoods of zero in E for tR. Since t is finer than tR' R is continuous, hence
E E-
~R.
Same argument for the other cases. REMARK. In prop. II.3.16, we oan obviously reduce U to the class of all l.c.s. equipped with the finest locally convex
to-
pology. Hence, for instance, prop. II.3.16 is still true when U is the class of barrelled spaces, ultrabornological spaces, etc. PROPOSITION Ilo3o17o [Mahowald, 7 ] . j ! 'iFis the class of all Banach spaces, led spaces.
~U ~
&u are equal to the class of all barrel-
Every barrelled space belongs to ~U , hence to &U • Indeed, let E be barrelled and R be a linear relation of E into a Banach space F such that ~(R)
E
X
= E and G(R) is closed in
F.
For every x E- E, if xRy, then xRy 1 ~ y
1
E- y + R(O)
•
Moreover, R(O) is a vector subspace of F and it is closed, since
22
{0}
R(O) = G(R) n ( {0} x R)
X
•
Thus R induces a linear map T of E into F/R(O), where F/R(O) is another Banach space, and G(T) has a closed graph, since G(T)
{0}
with
=
I ( {0}
G(R)
x R(O)
R(O))
X
'
C G(R).
By prop.I.2.3, T is thus continuous. Since, for every e C F, R- 1 (e) = T- 1 being the image of e in F/R(O)), R is thus
e (e
continuous. Conversely,
&~
is contained in the class of all barrelled
spaces. Let E belong to
Let E 9 be the space E equipped with the topology defined by the semi-norm p 9 , gauge &~and
9 be a barrel in E.
E9
of e, and
the completion of E 9 /ker p 9 • It is a Banach space. The canonical map j of E into E9 has a closed graph. Indeed, assume that (x,y) f
E•
of
Given co
> o,
G(j). Denote by
choose x' f
9
the unit ball
E such that jx' - y f
For each neighborhood U of zero in E, there exists z f
...
-
co
e.
E such
-
u and jz y f e:oe. Then jz jx' f 2c 9, 0 2e: oe 2 e: 0 e and (x-x 1 +U) n 2 e: 0 e :1= p. Thus X - x' f 2 e: 0 e and jx f jx' + 2 E S( y + 3e: 0 Therefrom jx = y and j has a clo0 sed granh. that x-z f
z
-
x' f
e.
By assumption, j
is thus continuous, hence 9 is a neighbor-
hood of zero in E and E is barrelled. *PROPOSITION II.3.18 [18]. ces,
~~and
&~are
1£
~is
the class of all Banach spa-
equal to the class of all ultrabornological
spaces.
We know that ~~ C &~. By prop. 11.2.4, ~~ contains all Banach spaces. By prop. 11.3.12, it also contains all ultrabornological spaces. Assume now that the l.c.s. E belongs to
&~.
Let 9 be an ul-
trabornivorous disk of E (cf. def. 111.2.1). Define the Banach space
E8
and the map j
as in the preceding proof. To see that 9 23
is a neighborhood of zero, we have to verify that j is c ontinuous or, since E E- gt that G( j) is fast sequentially closed. fF ' I f (xm' jx ) is fast convergent to (x,y), there is a fast m compact disk K such that xm- x in EK (prop. III.1.7)• Since 9 absorbs K, it follows that x m x for p 9 • Thus jx m - jx in E. . But jxm - y in E • Thus y = jx and G(j) is fast sequentially 9
closed. 4. THE RANGE SPACE DEFINITION II.4.1. Let us denote by
~E
the class of all spaces
F such that every linear relation R of E into F such that ~(R)
= E and G(R)
is closed in E x F is continuous. If f1 is a
class of l.c.s., denote by ~g the intersection of ~E (Ef&). The next two natural questions are a) to find good permanence properties for
~E
and
~g•
b) to characterize ~g for important classes &. The following results about question a) are essentially due to Raikov [87]. PROPOSITION II.4.2.
~
F, G be l.c.s. and T a continuous li-
near map of F ~ G. If F E- ~E' ~ G E- ~E. Let R be a linear relation of E into G. Then T- 1 o R is a linear relation of E into F. If the graph of R is closed in E x G, the graph of T
-1
oR is closed in E x G
let x , y be a a generalized sequences such that x - x in E, y - y in F and a a x (T- 1 oR)y (i.e. xaR(Ty )). Since Tis continuous, Ty - Ty, a a a a 1 thus xR(Ty) and x(T- oR)y. IfF E- ~E' T- 1 oR is thus continuous, hence R = ToT- 1 oR is continuous. COROLLARY II.4.3. 1£ Ft E- ~E' ~ Ft 1 E- ~E for every locally convex topology t' ~ t. 1£ F E- ~E , then F/N f ~E for every closed subspace N s£ F.
24
PROPOSITION II.4.4. l£ F f F belongs to
~E
, then every closed subspace G
~
~E.
Let R be a linear relation of E into G. It is of course a linear relation of E into F. Since E x G is closed in E x F, the graph of R is closed in E x F if it is closed in E x G. Finally, R is continuous of E into F if and only if it is continuous of
E into G. Hence the result. DEFINITION II.4.5.
Let
~
0
be the class of l.c.s. F such that
every linear relation of E into F such that
~(R)
is non meagre
in E and G(R) closed in E x F, is continuous of E into F and such that ~(R) = E. The class
~0
contains all
~-spaces.
It will be proved in a
more general context in prop. IV.5.3. It is clear that the above results (prop.II.4.2-3-4) hold true for
~
0•
PROPOSITION II.4.6. ( i E-m) Eo ~0 ~ F. ~
l£ '
F is the union of countably many subspa-
~
F Eo ~o·
Let R be a linear relation of E into F, such that ~(R) is 00 R- 1 F. is non meagre non meagre in E. Since ~(R) = u R- 1 F. ~ ~ ' i= 1 for some i E-JN. Consider then the restriction R 1 of R defined by xR'y
~>
xRy and y f
Fi
It is a new linear relation of E into F.
~
non meagre in E and G(R
1 )
, such that ~(R 1 ) is
is closed in E x F .• It is thus conti~
nuous. This yields the continuity of R for every nei~hborhood 1 1 V of zero in F, R- v ) R 1 - (vnF.) and R•- 1 (VnF.) is a neighbor~
~
hood of zero in E. Moreover ~(R) ) ~(R') =E. Hence the result. COROLLARY II.4.7. A countabl• inductive limit of elements of belongs to ~ •
~
0
Indeed, if F is the inductive limit of the spaces F. (i f:m), ~
it is the union of the images ·• Combine then prop. II.4.2 ~ 25
0
(extended to ~ ) and prop. II.4.6o 0 No further result seems to be known regarding finite or countable products or projective limits. If, instead of ~E , we consider the corresponding class defined by replacing linear relations by linear rraps, prop. II.4.2 requires the extra assumption that T be injective. So quotient spaces are lost. At this stage, we have two possibilities. The first is to characterize
~C
for suitable
c.
This question and related pro-
blems were solved for various classes of spaces, in terms of completeness or "Krein-Smulian" properties (cf. the theory of Ptak evoked in the comment of prop. I.2.3). We do not intend to develop that theory in the present notes. Let us however mention an interesting result of Komura which underlies much of the current
resear~
in this direction.
DEFINITION II.4.B. A class C of l.c.s. is called inductive if it contains all finite dimensional l.c.s. and is stable under ~nductive
limits (we take here the most general definition of
inductive limits of l.c.s., i.e. the finest locally convex topology on a given space E which makes a suitable collection of maps ranging into E continuous). Let (E,t) be a l.c.s. The associated €-topology, denoted t 8 or t 8 (E), is the coarsest topology on E finer than t and for which (E,t 8 ) is a member of c. The existence of t 8 is proved as follows. First E, equipped with its finest l.c. topology, is the inductive limit of its finite dimensional subspaces, hence a member of is the inductive limit of all (E,t
1 )
c.
Then (E,t 8 ) ~ C such that t 1 ~ t.
We will assume in the sequel of this paragraph that C is inductive. PROPOSITION II.4o9o
l!
T is a linear and continuous map of
(E,t) ~ (F,t•), it is still continuous of (E,t 8 ) ~ (F,t In particular, iLL is a subspace of (E,t), t 8 (L) is finer than the restriction to L of t 26
8 (E).
0).
Let t" be the inductive limit topology on F o£ (Et ,T) and &
(Ft 1 ,id) : i£ U (resp. r) is a base of neighborhoods o£ zero in &
E (resp. F), it admits {TU+V : U f U, V f r} as a base of neighborhoods of zero. Since & is inductive, (F,t") f &. Moreover t& ~ t, t~ ~ t 1 and Tt ~ t 1 ITE imply that t" ~ t•. Hence t" ~ tg and T is continuous of (E,t&) into (F,tJ;.). DEFINITION II.4.10. Denote by Br (&) the class of all spaces F such that every linear map T o£ E f & into F is continuous whenever its graph is closed. PROPOSITION II.4.11. (Komura, 61]. A l.c.s. (E,t) belongs to Br(&) if and only i£, for every l•C• topology t 1 ~ E such that t t t ~ = t ft" The condition is necessary. Indeed, i£ t 1 ~ t, the identical map i of (E,t 1 ) into (E,t) has a closed graph (its inverse is continuous). Thus i has a closed graph as a map o£ (E,tJ;.) into (E,t). Since (E,t) f Br(&) and (E,t') f &, i is thus continuous, t
I
~
hence tg ~ t. This yields tJ;. ~ t&. But tJ;. ~ t& , hence the result. The condition is sufficient. Let (F,t 1 ) belong to & and T be a linear map of (F,t 1 ) into (E,t), such that G(T) is closed. Denote by U (reap. r) a base of neighborhoods of zero in E (resp. F). The topology t" which admits {TU+V : U f U, V f r} as a base of neighborhoods of zero is locally convex and Hausdorff. Indeed, if y f TU + V, ((o,y) + U x V] n G(T) ~ p. Thus y f TU + V, VU f U, VV f r
l> (o,y) f l>
y =
To
'G""['T)
= G(T)
= o.
It is clear that t" ~ t. Thus t~ = t& ~ t. On the other hand, Tis continuous o£ (F,t 1 ) into (E,t"), hence o£ (F,t 1 ) into (E,tg)' by prop. II.4.9. Hence the result. A dual.characterization of Br(&) is easily obtained when & consists of Mackey spaces.
27
PROPOSITION II.4.12. Assume that & is contained in the class of all Mackey spaces. ~ (E,t) f
Br(&) if and only if every
vector subspace L ~ E* such that L n E 1 is a(E•,E)-dense in E 1 and that (E,~(E,L)) f
&, contains E'.
Here ~(E,L) denotes the topology of uniform convergence on 0
(L,E)-compact absolutely convex subsets of L. The condition is necessary. Take such a subspace L and consi-
der the identical map i that ~(i*) = L n E
1 •
of (E,~(E,L)) into (E,t). It is clear
So the density of L n E 1 in E' means, by s
prop. I.2.5., that i has a closed graph. It is thus continuous, hence ~(i*)
=
E 1 and L) E 1 •
The condition is sufficient. Let t' be coarser than t. The identical map i
of (E,t') into (E,t) has a closed graph. Thus,
by prop. I.2.5,
(Et
1 ) 1
g Et' is a Mackey space,
nEt is dense in E~. Moreover, since
g
(E,~(E,(Et 1 ) 1 )) g
Thus (Et' ) 1 g
)
=
(E,tg) E & •
E 1 and, since Et' is a Mackey space, t g
0~
t. The
conclusion follows by prop. II.4.11. We refer to the literature for further developments along this line. We have sketched above an approach to a characterization of the classes of range spaces. The descriptions of these classes turn out to be difficult to handle for practical purpose, i.e. when deciding, in concrete examples, wether a space belongs to them or not. Thus it may be more important to exhibit nice subclasses of ~&' large enough to contain those spaces that occur in practice and having reasonable permanence properties. We already mentionned that the set & of all ultrabornological
-
spaces would be an acceptable class of domain spaces. There do exist interesting subclasses of the corresponding
~g•
The first
one was introduced by Raikov [87]. Next L. Schwartz showed that contains all Souslin spaces. Finally, the author introduced the class of webbed spaces. The theory of these spaces will be
~g
developped in chap.IV,V,VI and the results of L. Schwartz in chap.Vn. 28
III Bomological and ultrabomological spaces
1. FAST CONVERGENT SEQUENCES To substantiate the idea that the class of ultrabornological spaces is adequatly comprehensive for the needs of applications, this chapter is devoted to their study. The strong analogy with bornological spaces makes it reasonable to handle both cases together. DEFINITION III.1.1.
Let E be a locally convex Hausdorff space.
An absolutely convex subset. A of E is also called a disk. If A is a disk of E, tA denotes the topology for which
{x + r.A : r. ) 0}
is a base of neighborhoods of x for ePch x E E. The linear hull of A equipped with the topology induced by tA is a semi-normed space, its topology being defined by the semi-norm pA , gauge of A. We denote it by EA.
If EA is normed, we say that A is
normin~
If it is moreover complete, A is a Banach disk (or a completing ~).
A disk A absorbs a subset e of E if e C AA for some A. LEMMA III.1.2. The disk A absorbs e if and only if, given a seguence An E- ]0,1] (nE-lN), A absorbs sequence x n
If A does not absorb e, r
n E lN } .:.f~o::.;r.._.:::.e~v..::e:..:r~yL.
{A n x n
(n E- lN) E- e. take xn in e "- 2
n
-1
An A. Then, for each
E-lN, Ar xr ~ 2rA, thus A does not absorb {Anxn : n E-lN}.
The converse is obvious. DEFINITION III.1.3. A sequence x (n flN) is Mackey convergent n
(reap. fast convergent) to x if there is a bounded disk (reap. a bounded Banach disk) B of E sue~ that xn -
x in EB. A Mackey
(or fast) convergent subsequence is clearly convergent to the
same limit. A subset K of E is fast compact if it is compact in EB for some Banach disk B. PROPOSITION III~1.4. .:.C:..:O~n::..v~e~x~h::..u=l~l:........:o~f:..
If x (n ~ m)- O, the closed absolutely n n ~ nq is the set of all convergent series
{xn
~ I en I :!!: 1. If all such series are n=1 n=1 convergent, co({xn: n ~ m}) is moreover compact and metrizable. 00
~
of the form
00
en xn .!!i.1h
Conversely, i£ E is metrizable, every precompact subset of E is contained in the closed absolutely convex hull of a sequence xn(n~m),
convergent to zero. 1\'!oreover, the sequence xn(n ~ JN)
can be choosen in a dense subspace of E. Consider the linear map T of 1 1 into the completion defined by Tc(n~m) n
~
C
n=1
X
n
E of
E
n
It is continuous from the unit ball B of 1 1 , equipped with the a(l 1 ,c 0
E.
norm p of ~
n=1 Given £ >
sup n:!!:N
en
o,
Indeed, take any continuous semi-
We have
00
p(
E.
topology, into
)
X
n
-
N
00
~
n=1
CI n
<
sup p(xn) n>N
I en -c~ I < £/2 [
X
n
) :!!:
lcn-c~l p(x n ) + 2 sup p ( xn) •
~
n=1
n>N
£/4 for N large enough, thus
N
00
~ n=1
p(x )+1] n
=>
p( ~
C
n=1
n
X
n
~
n=1
c'n x n )<£·
Since B is equicontinuous in 1 1 ~ c~ and c 0 separable, B is compact and metrizable for a(l 1 ,c 0 ). Thus K = TB is compact in E. Moreover the topology t
of K is the
i~age
by T of the topology
0 (1 1 ,c 0 ) of B. The latter has a countable base hence t has also a countable base and is thus metrizable.
-
On the other hand K is absolutely convex and closed and, i f
C o ({xn
n ~ m}) is the closed absolutely convex hull of
{xn : n ~IN} in
E,
-
we have
{xn : n ~ IN} C K C Co ( {xn
30
n
E-JN})'
hence
,.... K
Co({xn: n E-lN})
The closed absolutely convex hull 'C";( {x -
fxn : n E- IN}
-
in E is equal to Co ( {x co
con
the set of all series E c x ( E . · n= 1 n n n= 1 vergent J.n E.
: n E- lN}) of n
: n E- lN}) jc
n
n E,
thus it is
I~ 1) which are con-
Assume now that E is metrizable,
that D is a dense subspace
of E and that K is precompact in E.
Let Un(n E- IN) be such that
2n+ 1 un(n E- IN) is a base of decreasing absolutely convex neighborhoods of zero in E. There exists a finite covering of K by
t
translates
xi + u 1 (i:!EOn 1 ) of u 1 , with xi E- D. Define Ai(x) (i:!EOn 1 , x E- K) by Ai(x) = 1/2 for the smallest i such that x E- xi+ u 1 and Ai(x) = 0 otherwise. Then
n,
K1 = {x- _E Ai(x)xi : x E- K} l.=1
is a new precompact subset of E, contained in u 1 • Proceed to the same construction for K 1 and n2 .
1
(- x 1. + u2 ) 4
and Ai(x) (n 1
>
define
1
u
l.=n1+
lest i
u2
n 4U 1 such that
xi(n 1 (i:!EOn 2 ) E-D
K) such that Ai(x) = 1/4 for the smal-
n 1 such that n1
X-
E
j=1
1 A.(x)x. E- -4
J
J
X.
l.
+ U2
and A·(x) = 0 otherwise. l. We obtain, by induction a sequence x.(i E- lN) contained in D l. and convergent to zero and sequences Ai(x) (i E- lN) (x E- K) such that co
x =
co
E Ai(x)xi and E i=, i= 1
hence the result.
jAi(x)j ~ 1, ~x E- K,
PROPOSITION III.1.5. The closed absolutely convex hull of a fast convergent sequence is fast compact. Conversely, a fast compact set is contained in the closed absolutely convex hull of a fast convergent sequence. Assume that x (n n
~m)
is fast convergent, thus convergent to
zero in EA for some Banach disk A. By the preceding result,
"Cc;'( {xn : n ~JN}) (closure in EA) is compact in EA' thus fast compact in E. It is thus compact, hence closed in E and therefore coincides with the closed absolutely convex hull of {xn : n ~ JN} in E. Conversely, if K is fast compact, it is compact in some Banach space EA. By the preceding result, there exists then a sequence xn(n E-m)- 0 in EA such that K C "Cc;'( {xn : n ~ JN} ). COROLLARY III.1.6. The closed absolutely convex hull of a fast compact set is fast compact. Obvious. PROPOSITION III.1.7. I f a set K (or a sequence x (m ~m)) .i.!!, m fast compact (or fast convergent to x), there exists a fast compact disk K 1 such that K (..Q.!:. xm(m ~ m)) is fast compact (..Q.!:. fast convergent to x) ~ EK,. Let x (m E-m) be fast convergent to x. There exists a bounm ded Banach disk A such that xm - x in EA. Thus xm - x ~ £mA' with £m- 0. Choose £m
>
0. If Am
= 1/V£m' we have
Am(xm-x) E- V£m A thus Am(xm-x)- 0 in EA. The sequence y 0 y m =A m(x m-x)(~1) is fast convergent to zero, hence K1
"Cc;' ({y: m m~O}) is fast compact in E. Moreover, x and
xm(m
~m) ~
~
r-
K•, thus xm- x in EK•" m Let K be fast compact. Then K C Co ( {ym: m ~m}), where ym is fast convergent to zero in E. By the first part of the proof, it is fast convergent to zero in EK, for some fast compact disk K'. Since the closed absolutely convex hulls of {y : m e-m} in --m E and in EK, are the same, Co( {ym : m ~IN}) is fast compact in
32
EK' and xm- x
x,
EK'' hence K is fast compact in EK'" PROPOSITION III.1.8. set B
~
E,
1£
E is metrizable, for every bounded sub-
there exists a bounded disk B' such that the topolo-
cies induced by E and by EB, coincide on B. Let U (m fIN) be a base of absolutely convex neighborhoods m ~ of zero in E. There exist Am > 0 (m flN) such that B C n X u • m m m=1 00
n
The set B 1
2mAm Urn is bounded in E. Voreover, for each
m=1 x f
B and
E
>
O,
N
(x +
E
n m=1
2mx
u ) n m m
B
c
x + (B-B)
n
N E
n
m=1
um
00
(
X
+
n
E
2m Am urn
m=1
PROPOSITION III.1.9. If a bounded disk of E is sequentially complete, it is completing. Let B be a sequentially complete bounded disk of E. Let xm(m flN) be a Cauchy sequence of EB. Taking a subsequence, we -m may assume that xm- xm_ 1 f 2 B for each m flN. Then
x 1 + (x 2 -x 1 ) + ••• + (xm-xm_ 1 )
xm f
x 1 + 2- 2 B + ••• + 2-mB C x 1 + B, lfm f lN •
Moreover, xm(m f lN) is a Cauchy sequence in E. Thus it converges for the topology of E to some x f
x 1 + B C EB. For each m0 f lN,
by the above argument, xm - x
2
-m
mo
complete, we have thus x - x f m0 PROPOSITION III.1.10.
1£
E is an
f
°B. Since B is sequentially
-m
2
°B, hence x m -
~-space,
x in EB.
its compact subsets
are fast compact and its convergent sequences are fast convergent. If K is compact in E, it is bounded hence, by prop.III.1.8, there exists a bounded disk B such that the topologies induced
33
by tB and by E coincide on K. Thus K is compact in EB. By prop. 111.1.9, B is a Banach disk. Since tB ~ tB' K is still compact in EB. PROPOSITION III.1.11. ~ E,F be l.c.s~ If a linear map T ££ E into F maps fast convergent sequences of E into bounded sequences of F, it maps bounded Banach disks into bounded Banach disk~ fast compact sets into fast compact sets and fast convergent sequences into fast convergent sequences. Let B be a bounded Banach disk of E, By lemma III.1.2, TB is bounded if {A n Tx n : n ~ m} is bounded for all X n ~ B and An- 0. But An x n (n ~ m) is fast convergent, thus An Tx n (n ~lN) is bounded. Hence TB is bounded. To show that TB is completing, take a Cauchy sequence xn(n ~ m) for the topology tTB. It has a subsequence such that x sequence y
mk+1-
mk
~
- x E (k
mk ~
(k ~ m) mk ~ 2-k TB for each K. There exists then a
m) such that Ty
~ 2-kB. Thus ym
mk
= x
mk
X
and
converges for tB to some k
thus x
mk
-
Ty for tTB and xm -
Ty for tTB•
The map T is of course continuous from EB to FTB• Thus, if K is compact in EB' TK is compact in FTB' hence fast compact iD F.
Same argument for fast convergent sequences. 2. BORNOLOGICAL AND ULTRABORNOLOGICAL SPACES DEFINITION III.2.1. A disk A C E is absorbing, bornivorous or ultrabornivorous if it absorbs every element, every bounded disk or every bounded Banach disk of E.
By lemma III.1.2, the disk A is bornivorous or ultraborniJLorous if and only if it absorbs every Mackey convergent or Last convergent sequence of E. Thus, for instance, A is ultrabornivorous if and only if it absorbs the compact disks of E. Indeed, if xm(m E-lN) is fast convergent, ~( {xm 1m E- lN}) is fast compact, hence compact. Thus, if A absorbs the compact disks, it also absorbs the fast convergent sequences. A straightforward improvment consists in considering only
various kinds of "quickly decreasing sequences" (i.e. sequences of the form Am x m, with x m E- B bounded (and completing) and Am decreasing fast enough~ For further details, see [47,48]. The space E is bornological (of ultrabornological) if every bornivorous (ultrabornivorous) disk is a neighborhood of zero. PROPOSITION III.2.2. The space E is bornological (resp. ultrabornological) if and only if it is the inductive limit of the spaces EB (B bounded disk (reap. bounded Banach disk) ~E). A disk A is a neighborhood of zero in the inductive limit topology if and only if it contains
a neighborhood of zero of
each EB' i.e. a homothetic of B for each B, hence the
conclusio~
It is also clear that if E is bornological, it is the inductive limit of the spaces EB' where the B 1 s are the closed absolutely convex hulls of the Mackey convergent sequences, or even of more quickly decreasing sequences (see the above remark). It is not known whether a minimal family of such bounded subsets exists. DEFINITION III.2.3. Let t be the topology of E. The associated bornological (reap. ultrabornological) topology, denoted by tB (reap. tub) is the coarsert locally convex topology finer than t for which E is bornological (reap. ultrabornological). We usually write Eb (reap. Eub) for (E,tb) [reap. (E,tub)].
PROPOSITION III.2.4. The space Eb (resp. Eub) is the inductive limit of the spaces EB (B~~) where ~ is the class of all bounded disks,
or of the closed absolutely convex hulls of all sequences
'
which are Mackey convergent to zero (resp. where ~ is the class
of all bounded Banach disks, or of all fast compact disks of E), In particular, E ~ Eb (resp. Eub) have the same bounded subsets (resp. bounded Banach disks). Thus the set of all bornivorous (resp. ultrabornivorous disks of (E,t) is a base of neighborhoods of zero of (E,tb) (resp. (E,tub)). Call t 1 the topology of the inductive limit quoted in the assertion. Of course t' is finer than t. Thus its bounded subsets (resp. its bounded Banach disks) are again bounded subsets \resp. bounded Banach disks) of (E,t). It follows moreover from the definition or from lemma III.1.2 that the bounded subsets (resp. the bounded Banach disks) of (E,t) are bounded in (E,t
1 ).
In the latter case, they
are even completing, by prop. III.1.11. By prop. III.2.2, (E,t') is bornological (resp. ultrabornological). Moreover, if (E,t") is bornological (resp. ultrabornological) and t"
~
(E,t
is bornivorous (resp. ultrabornivorous) in (E,t), hence
1 )
t, every absolutely convex neighborhood of zero in
in (E,t"), thus it is a neighborhood of zero in (E,t") and t'
~
t". Hence the result.
EXAMPLE III.2.5 • .&!U, E be a l.c.s. _The space (E~(E' ,E))ub.!.!. the inductive limit of the spaces E 1 00 (U~U), where U is a base of neighborhoods of zero for the Mackey topology of E. Indeed, if B is a fast compact disk in E 1 for a(E 1 ,E), it is absolutely convex and compact for a(E 1 ,E), hence contained in some
uo.
E~(E 1 ,E)"
~6
Conversely, every 0° is a bounded Banach disk of
PROPOSITION III.2.6. If a linear map of E
~
F maps bounded
subsets or Mackey convergent sequences (resp. completing bounded subsets or fast convergent seguences) of E into bounded subsets of F, ~ T is continuous of Eb (reap. Eub) ~ Fb" Replacing "bounded subsets of F" ~ "completing bounded subsets of F", we get that T is continuous of Eb (resp. Eub) ~ Eub" Indeed, if U is a neighborhood of zero in F, T- 1 u is bornivorous (or ultrabornivorous), hence a neighborhood of zero in Eb (resp. Eub). Observe next that F and Fb (or F and Fub) have the same bounded (bounded and completing) subsets. In particular, ~ E is bornological (or ultrabornological), under the aame assumptions, T is continuous of E DEFINITTON III.2.7.
~F.
Let us denote by ~me (resp. ~fc) the set
of all sequences x m(m flli), Mackey (reap. fast) convergent to zero, and by E' me (resp. E} 0 ) the dual equipped with the correspending topology t~
(resp. t~ ). me fc By prop. III.1.5, the topology of E} 0 is also the topology
of uniform convergence on the fast compact subsets of E. A linear form on E is locally bounded (resp. fast bounded) if it is bounded on the bounded (resp. bounded Banach) disks of E. The set of all locally bounded (fast bounded) linear forms on E is clearly the dual of Eb (resp. Eub). PROPOSITION III.2.8. The space (Eub)' is the completion of E£ 0
•
The space (Eb)• is contained in the completion of E'me • Let ~·me (resp. ~f' c ) denote the set of the closed absolutely convex hulls of all sequences x m (m f m) Mackey (reap. fast) convergent to zero. In the second case, it is simply the set of all fast compact disks (prop.III.1.5). Recall [62, p.270] that the completion of E~c (resp. E£ 0 ) is the set of all linear forms onE, cr(E,E 1 )-continuous on each element of ~~c (reap. ~£ 0 ). Let y belong to the completion of E£c and K be a fast compact disks of E. It is also cr(E,E 1 )-compact, thus y is bounded 37
on K, hence fast bounded. Conversely, let y be fast bounded and K be fast compact. By prop. III.1.7, there exists another fast compact disk
K'
such that K is compact for tK,. The linear form y is bounded on K', hence continuous for tK,. Moreover, tK' and a(E,E') are equal on K, hence y is a(E,E 1 )-continuous on K and it belongs to the completion of E£ 0
•
Let now y be locally bounded and B E- JJ 1 : B = ~( {x : m E-lN}), me m where x is Mackey convergent to zero. There is another B' E- JJ' m me such that x m - 0 in EB 1 : take B 1 = C";;( {X m x m : m E- lN}) where
Am - ~ but Am x m(m E- lN) is still Mackey convergent to zero. The gauge semi-norm of Bt is pB 1 (x) =
sup x 1 E-B 10
l<x,x'>l
•
The sequence xm(m E-lN) is precompact for pB 1 hence [89, p.51 precomp~ct
43, p.200], B 1 is sup x 1 E-{xm:m E-lN}
l<x,x'>l o
or
for sup x 1 E-B 0
l<x,x'>l
and, using once more the same argument, B is precompact for pB,. Since moreover B 1 is closed and absolutely convex in Ea(E,E•)' [43, p.95 or 62, p.395], tB, and a(E,E 1 ) coincide on B. Since y is bounded on B 1 , it is continuous for tB'' hence continuous for a(E,E 1 ) on B. Hence the proof. PROPOSITION III.2.9. The space E is bornological (resp. ultrabornological) if and only if it is a Mackey space and E' me (resp. E£ 0 ) is complete. Prop. III.2.9 and the main ideas in the proof of prop. III.2.8 are due to Kothe in the bornological case[62, 63]. If E is ultrabornological, it is bornological. If it is bornological, since the neighborhoods of zero for the Mackey topology ~ of E are bornivorous (Mackey boundedness theorem), E is a Mackey space. By prop. III.2.8, if E is ultrabornological, E£ 0 is complete, since (Eub)' 38
= E 1 • If E is bornological,
E~ 0 is complete.
Indeed, let
~
be a Cauchy filter in E' • It converges for me 0 (E*,E) to some y' f E*. For every sequence x (m f m) Mackey o m convergent to zero, there exists y' f E' such that sup l<xm,y~-y'>l ~ 1 • m Thus
y~
is locally bounded, hence continuous.
Conversely, if E~c (resp. E}c) is complete, then {Eb)' {resp. (Eub)' {resp. E
= E1
= E 1 ) . If E is moreover a Mackey space, E = Eb
= Eub).
3. PERMANENCE PROPERTIES AND EXAMPLES Let us first mention an easy connection between bornological and ultrabornological spaces. PROPOSITION III.3.1. l£ E is bornological and sequentially complete, it is ultrabornological. Indeed, every bounded disk is contained in its closure which, by prop. III.1.9, is completing. PROPOSITION III.3.2.
~
T be a linear map of E
~
F.
1£
E
is bornological (ultrabornological) and if T is open and maps Mackey convergent {resp. fast convergent) sequences into bounded sequences, ~ F is bornological {resp. ultrabornological). If A is a bornivorous (resp. ultrabornivorous) disk of F, 1 T- A is bornivorous (resp. ultrabornivorous) in E (for the latter case, apply prop. III.1.11). It is thus a neighborhood of zero in E, hence A is a neighborhood of zero in F. PROPOSITION III.3.3. The inductive limit of bornological (ultrabornological) spaces is again bornological (ultrabornological). Let E be the inductive limit of the spaces E and T be the canonical map of E to E. If A is bornivorous i~ E, T-~A is a a bornivorous in E • If E is bornological, it is thus a neigha
borhood of zero in E and this for all a implies that A is a a neighborhood of zero in E.
39
Same proof for the ultrabornological case. REMARK. Since it is obvious that finite dimensional l.c.s. are bornological and ultrabornological, we have thus proved that the class of all bornological (resp. ultrabornological) is inductive (cf. def.II.4.8). PROPOSITION III.3.4. ~ F be a subspace of (E,t) such that every locally (resp. ~) bounded linear form on E vanishing ~ F is identically zero. l£ F is bornological (resp. ultrabornological),so is E. By use of the Hahn-Banach theorem, F is dense in Eb (resp. Eub). By prop. II.4.9, we have, in F,
By assumption tb(F) = t on F, thus tb(E) = t on F and, due to the density of F in (E,tb), tb = t on E. Same argument for tub• PROPOSITION III.3.5. The space mX (~eX) is bornological and ultrabornological if card X is smaller than the first strongly inaccessible cardinal. A cardinal a is said to be strongly inaccessible if (i) a (ii) ~ (iii)
> <
~ y~r
0
'
a~
~y
<
2~
<
a
a if ~y
<
a for each y
~
r and card r
<
a.
We refer to [62) for the proof that EX or eX is bornological. SincemX and CX are moreover complete, they are then ultrabornological. PROPOSITION III.3.6. (Mackey-Ulam). The product of a bornelogical (ultrabornological) spaces is bornological (ultrabornological) iL a is smaller than the first strongly inaccessible cardinal. We will prove a somewhat more general result.
40
pROPOSITION III.3.7. If card V is less than the first strongly inaccessible cardinal,
( n
~
E. )b = ~
iE-V
It is clear that the Nackey-Ulam theorem is a particular case ot the above result. Let tb be the topology of (
n
iE-V
E, )b. By prop.III,2.4, it ...
admits the bornivorous disks as a base of neighborhoods of zero. By prop.III.3.5, the topology induced by tb in each simple subspace of ced by
n
iE-V
n
iE-V
(E.)b is thus coarser than the topology indu~
(E. )b. The same thing is obviously true for the fac~
tor spaces. Thus, by cor. II.3.8, the tonology of ( n E, )b is iE-V ... coarser than the topology of
n iE-V
(E. )b. 1
Conversely, the sets
where v C V is finite and each U.l. bornivorous in E.l. are clearly
n E. and this yields the converse. iE-V 1 Same proof for the ultrabornological case.
bornivorous in
There are few informations on the subspaces of a bornological (ultrabornological) space. Of course, a complemented subspace of a bornological (ultrabornological) space E is bornological (ultrabornological).
Indeed, it is isomorphic to a quotient of
E. Apply then prop. III.3.
PROPOSITION III.3.8. subsp~ce
[Dieudonne, 29]. A finite-codimensional
L of a space E is bornological if E is bornologicalo
More generally, Lb has the topology induced by Eb. It is clearly enough to consider a subspace L of codimension one, thus the kernel of a linear form
x•.
Let
E- E be 41
such that (x
0
,x'> = 1 and denote by T the projection
x - x - <x,x'>x 0
of E onto L.
Assume first that x'
is locally bounded. Then T maps bounded
sets of E into bounded sets of L, If 9 is a bornivorous disk of L, T- 1 9 is a bornivorous disk of E and 9 = L n T- 1 e. If 9 1 is a bornivorous disk of E, 9
1
n
L is bornivorous in L. Hence Lb has If E is bornological,
the topology induced by Eb.
L is thus bor-
nological too. Assume now that x' bounded sequence x
~
X
m
is not locally bounded. There is then a
m such that <x ~ ,x'> = m for each m. From
1
m -= -m Tx m + x o 1 - Tx m m
we deduce that Ym
X
the absolutely convex hull of
X
0
in the space EB , where B 0 is
0
and x ( m E- lN), m
0
thus a bounded
disk of E. Let 9 be a bornivorous disk of L. Denote by
~
the family
all bounded subsets of E containing B • For each B exists CB > 0 such that CB B
n
L C
e.
0
~
~.
of
there
Define 9
=
1
C~B)
Co( U
•
BE-~
It is of course
a
bornivorous disk of E. Moreover,
e' n 1 c 2e • Indeed, if
X
X
E- 9
1
( *)
n
L, we have
N !:
N !:
i=1
i=1
N
ai T z i
+
E i=1
a 1.
x 0
with N E
!ail~ 1, zi ~ cB.
i=1 since x
Bi' lfi, and
i=1
l.
~
N E
L. Therefore, N
X
= E i=1
a.(Tz.+y ), l. l. l. m
lfm
(the coefficient of y m is zero!). But, for each i,
42
0
z.l. + (z.l. ,x'>(y.m -x o )
~ CB for m large enough, hence
1
B.1. + ~ (z.l. ,x 1 )B o C 2CB. Bi C 29 m 1
(*).
Take now 1
9 " = c o ( 9 u 29 ' ) ;
n
it is a bornivorous disk of E, such that 9"
L =
e.
It is obvious that a bornivorous disk of E induces a bornivorous disk of L, thus we have proved that Lb has the topology induced by E 0 • This implies as above that L is bornological if E is bornological. See [110] for the ultrabornological case. Let us now turn to criteria for dual spaces to be bornological. As far as applications are concerned, the main result is the following, due to L. Schwartz [97] when E is a Schwartz space and improved in the present form by Hogbe-Nlend [48]. DEFINITION III.3.9. The space E is called an infra-Schwartz space if, for every neighborhood U of zero, there is an absolutely convex neighborhood V of zero, such that U0 is weakly compact in the Banach space generated by
vo.
PROPOSITION III.3.i0. The strong dual of a complete infraSchwartz space is ultrabornological. Let E be an infra-Schwartz complete space. Denote by b(E•,E) the strong topology of E 1 and by t the topology of the inductive limit of the Banach spaces E
00
(U neighborhood of zero in E).
Since every U0 is bounded for b(E 1 ,E), it is clear that t ~ b (E 1 ,E). Let us prove that (Et)' =E. This will imply-that t i s coarser than the Mackey topology of E 1
,
hence coarser than
b(E•,E). Therefore Eb(E•,E) will be equal to Et' hence ultr~ bornological. Since E is complete, by Grothendieck's completion theorem, a linear form on E' belongs to E (up to obvious identifications) if and only if it is continuous on each equicontinuous subset
43
of E' for the topology a(E•,E). Assume that x' ~ (Et)'. Given a neighborhood U of zero, there is another one V such that U 0 is weakly compact in Evo• The form x' is bounded over vo, hence continuous in Evo for the norm of Evo• It is thus continuous for the weak topology of Evo• Since uo is weakly compact in Evo' the weak topology of Evo is equal on uo to a(E•,E). Thus x' is a(E•,E)-continuous on U0
Hence the conclusion.
•
A converse to prop. III.3.10, identifying every ultrabornological space to the strong dual of a complete nuclear space, can be found in [48]. The following results are due to Grothendieck. DEFINITION III.3.11. A space E is called evaluable
if every
closed bornivorous disk is a neighborhood of zero. PROPOSITION III.3.12.
liE
is metrizable, the strong dual of
E is ultrabornological if and only if it is evaluable. First, if a space is ultrabornological, it is of course evaluable.
m)
Let Ui(i ~
be a base of absolutely convex neighborhoods
>
of zero of E. For every sequence A. in
Eb
~
0 (i ~
m), the closure
of co
9 =Co(
U Ai U~) i=1 ~
(*)
and its algebraic closure are equal. Recall that the algebraic closure
A of
a disk A is
n£ >0 (1+c)A.
It is clear that
To prove the converse, observe that, since each N
Di
a c 9.
is absolute-
ly convex and a(E•,E)-compact, Co( U A. U~) is also a(E 1 ,E)i= 1 ~ ~ compact [43, p.91], thus
N co( u
u~) ~ ~
A·
i=1
Take x' each N,
44
f
a.
N
= ( n .L Ai i=1
U.; ...
)o
There is some £ )
•
0 such that x'
f
(1+£)9. For
m)
sequence xn(n ~
N L
n ui such that <xN,x'> > 1+E. The i= 1 A.i is bounded. If w is its polar in E•, we have
hence there exists XN f
(x 1 +Ew) n 9 = ¢ • Indeed, otherwise there would exist N f N
u
Co(
y' f
i=1 such that x'
-
m
and
N
= ( n L u.l. ) 0 A.i U'?) l. i=1 A.i
y'
EW• How ever
~
hence a contradiction.
Eb
Assume now that nivorous disk in E
1 •
is evaluable and that 9
Since each U'? is a Banach disk, 9' conJ.
tains some 9 of the form (*). Then 29 1
)
e=e
e is a closed and bornivorous disk in
Since
bounded subset of
Eb
is an ultrabor-
1
is equicontinuous),
Eb
a is
(closure in
Eb).
(because every a neighborhood
of zero, hence the conclusion.
l!
E is metrizable and Eb separable, then is ultrabornological. By the proof of prop.III.3.12, we only need to check that
PROPOSITION III.3.13.
Eb
co
9
=
Co ( U A.. U'?) i=1 l. l.
(closure in Eb; A.i > 0) is a neighborhood of zero. Since Eb is separable, E' \ 9 is also separable for the topology of
Eb
(because it is open). Let x~(m ~m) be a countable
dense subset of E 1 \9 • For every m, x' 1 e, hence m 1 m m X 1 1 ( n -- U. ) 0 and there exists X ~ n l_ Ul.. SUCh that m i=1 "-i J. m i=1 "-i (x ,x'> > 1. The sequence x (m ~IN) is bounded in E. If w is . m m m 1 1 J.ts polar, 2w C 9. Indeed, if there exists x' E (2w)\ 9, by density, there exists i such that
45
supj<xm,x'-xi>J < 2 m
Then < J<xi,xi_>J <sup
J<xm,x'-xj_>J +sup
m
J<xm,x'>l < 1
m
hence a contradiction. COROLLARY III.3.14.
1£
E is metrizable, separable and semi-
reflexive, Eb is ultrabornological. ~
It is true in particular
E is a metrizable Schwartz space.
Indeed, by definition of semi-reflexivity, every bounded subset of E is contained in a weakly compact subset. Hence E b' -- E ,I · Since E is separable, E' is separable, hence E' is separas 't ble. Apply then prop.III.3.13.
One further step in the study of conditions for Eb to be bornological or ultrabornological consists in considering the dual of an inductive limit. The following result slightly extends a theorem of Grothendieck [6]. PROPOSITION III.3.15.
~
E be the strict inductive limit of a
sequence of evaluable spaces En (n ~ m). If each E'n, b is bornological, Eb is ultrabornological. Moreover, every bounded subset of E is contained in the closure of a bounded subset of ~ En•
REMARK. Since E n is evaluable, every bounded subset of E'n,b is equicontinuous, hence its closure is ~omplete. Thus E~ is quasicomplete and it is simultaneously bornological and ultrabornological. The same is true for E•. Denote by b 1 the topology of uniform convergence on the bounded subsets of the various En •s. We shall first prove that Eb' is bornological. Let
46
e
be a bornivorous disk in Eb,. There exists
~lN
such that
E~
E- e.
Otherwise, for each n, choose
x~
E- E~ \ e
0
hence
vanishing on E n • It is obvious that the sequence nx'n converges to zero in Eb'" Thus it is a bounded subset of Eb' and it is absorbed by e there exists C > 0 such that n
X I
n
lfn E- JN
€- C e t
which contradicts x~ Denote by en
f
t
e for n
> c.
the set of all linear forms on En 0
which can 0
be extended to E by an element of e. It is a bornivorous disk of E' b" Indeed, if B is bounded in E' b' since En is evano' no' o luable, it is equicontinuous. Since the inductive limit is strict, it is also the restriction to E
n
subset B 1 of E
For some C
1 •
> o,
of an equicontinuous 0
B 1 C C e. Thus B
c c en
0
The spaces E'n, b are assumed to be bornological. Hence b : there is some bounded en is a neighborhood of zero in E' o n no' 0 0 (polar in E~ ). set B in En such that en ) B 0
0
If now B 0 is the polar of Bin E 1 restricted to En 0
0 ,
for every x' E- B 0
coincides with the restriction to E n
,
x' of an
0
element of e. Thus B° C e + E° n
hence
t~e
C 2e 0
conclusion.
It is obvious that the topology b is finer than b 1 •
Let us
prove that they have the same bounded sets. It will imply that the absolutely convex neighborhoods of zero in Eb are bornivorous in Eb'' hence are neighborhoods of zero in Eb' and therefore b' = b. If B is bounded in Eb'' its restriction to En is bounded in E'n, b' thus equicontinuous for each n E-m. Therefore B is equicontinuous onE hence bounded in E'. Let finally B be bounded in E. ~ts polar is a neighborhood of zero in Eb' = Eb. Hence there exists a bounded subset B 1 of some E n such that B 0 ) B 10 Therefrom BC'C'O B', hence the last assertion of prop.I.).16.
47
IV Webbed spaces
1. WEBS DEFINITION IV.1.1. Denote by I the set { ( n 1 , ••• , nk)
k, n 1 , ••• , nk E- IN J
&
ordered by the following order relation a (m 1 , ••• ,mk) ~ (n 1 , ••• ,nk,) Let E be a l•c.s. A
~ ~
<
~ k ~ k'
and mi
of E is a map
=
n. ( i~k). l.
of I into the
set of all subsets of E such that
(a) efJ
E
=
(b) ~ ~ v
(c) e
, ~ e
U v>~
~
The web each e E-
~
~
~
)
e ,
v
e 't~
v
,
E- I •
is called convex, absolutely convex or closed if
is convex, absolutely convex or closed.
EXAMPLE IV.1.2. The webs usually occur in the following setting. Let e(k) (n,k E-m) be a collection of subsets of E such that, for e:ch k,
e~k)
(n E-m) is a covering of E.
We obtain a web by defining e
fJ
=
E and e
(n1, ••• ,nk)
=
e ( 1 ) n ••• n n1
DEFINITION IV.1.3. A completing sequence Bn(nE-lN) of a l.c.s. E is a decreasing sequence of subsets of E such that there exists a sequence A. n (n e-m) of non negative numbers such that countably many of them differ from zero and that 00
48
converges in E for all ~ n E [o,A n ] and x n E Bn • The sequence A (n EJN) is said to be associated to the sequence B (n EJN). n
n
The sequence B (n EJN) is strictly completing if the associan ted sequence A ( n E IN ) can be c h o o sen in such a way that n
co
L:
n=n +1
~n xn
E B
no
0
for all n 0 E IN, 1-Ln E [O,An] and xn E Bn. We say then that A ( n E IN)
is strictly associated t o B ( n E JN ) • n
n
DEFINITION IV.1.4.
Let :7r. = {e
v
:
v E I}
be a web of a l.c.s. E.
(n EJN) of elements of :7r. such that vn+ 1 > v n for vn all n E JN is called a strand of the web. So a strand is a se-
A
sequence e
quence e( ki i A
n1, ••• ,nk.
)(i EIN), where nk(k EJN) is arbitrary and
~
co.
web is said to be completing if each of its strands is
completing. It is strict if it is absolutely convex and if each of its strands is strictly completing. The space E is said to be webbed or strictly webbed if it admits a completing or a strict web. The following easy remarks are quoted for later reference. PROPOSITION IV.1.5.
If a l•c.s. E admits a completing
it
web~,
admits another completing web which elements are starshaped around zero. The new web ::R. 1 constituted by the subsets e'
v
=
u
H[0,1]
Ae
v
(e
v
E::R.)
is obviously completing if :7r. is. PROPOSITION IV.1.6. For each corr.pletin~ (resp. strict)~ ~
E,
there is another one ::Tr.'
such that each subset of E absor-
bed by the elements of a strand of :7r. is contained in the elements of a strand of The new web ::R.'
::rr.•.
is defined as follows
49
e'
n1
= m e (m 1 ,n 1 E-lN) 1 1
n
indexing (m 1 ,n 1 ) be a single index n1 E-m. Next, indexing (m 2 ,n 2 ) by n2 E- IN, e n1 1 ,n , = m1 e n n m2 e n ,n 1 2 1 2 1 and so on. We clearly obtain a new web.
=
A
strand of
~1
reads
n m2e n , n n ••• n mk i e n , ••• , nk. (mk,nk E-m for n1 1 1 2 each k; ki i oo), If A.i(i E-lN) is associated 1 or strictly associated to the strand e (i e-m) of ~, then n 1 , ••• , nk
&i
m,e
i
>..! J.
= A· / J.
sup mk
~k.
J.
is associated (or strictly associated) to the sequence &i (i E-:N), Comment. Completing and strict webs were introduced in [14], "r~seaux
under the name of
de type
~
et
r~seaux
stricts". We
use here the terminology of [89] and [90], although with a slightly different definition. In [14], it is assumed, in the definition of "completing webs", that A.k
.f=o
for each k E-lN. The slight improvment assu-
ming that only a subsequence does not vanish was introduced by M. Nakamura
[76].
Although bringing some technical troubles
later, it does not affect the properties of the webs and turns out to be useful for a topological interpretation of the completing webs. We had- however to change a little bit Nakamura's condition to preserve the stability with respect to products and projective limits. On the other hand, in condition (c) of def.IV.1,1, it would not make any difference to assume that U e is absorbent in v )j.l.
e
ll
v
(instea4 of being equal toe ). Prop.IV.1.5 would then proll
duce another web verifying (c) in its initial form. Various other types of webs are also introduced in [14]. M. Nakamura has proved the equivalence of several of them in
[76]. 50
The next elementary properties of completing sequences will be useful later. PROPOSITION IV.1.7. ~ Bn(nE-lN) be a completing sequence of E, A (n E-JN) an associated sequence of numbers and U a neighborn hood of zero in E. ~ k large enough, 00
{ ~ ~.x. i=k ~ ~
: ~i E- [O,Ai], xi fBi}
In particular, A.iBi C U
£££
~
i
C U •
k.
It is not a restriction to assume that U is closed. So it is enough to prove that n
{ ~ ~i xi : n ~ k, i=k
~i E- [O,A.i], xi f
for k large enough. If it is not the case,
Bi}
C U
there exist, by in-
duction, sequences mk,nk Too, ~i E- [O,Ai] and xi fBi such that nk
<
(mk~i~nk)
mk+ 1 and
nk ~
~·
~
x.~ ,
u,
'1k E- lN
This is impossible, since
~
• ~i
xi is convergent in E.
PROPOSITION IV.1.8. The sequence
B (n E-lN) is completing if n
there exists a sequence An (n ElN) of non negative numbers such that countably many of them differ from zero and that
is contained in a completing bounded disk for every sequence xn E- Bn(n E-lN)o Indeed, the sequence 2-n A (n E-JN) is then associated to n
Bn(n E-lN). PROPOSITION IV.1.9. ~ Bn(n E-lN) be a completing sequence. There exists an associated sequence An(n E-lN) such that
51
00
E lln xn .!!.W!, lln xn ( n flN) n=1 are fast convergent for all lln f [O,An] .!!.W!, xn ~ Bn• The idea of this result is due to M.H. Powell [83]. Let An(n flN) be a sequence associated to Bn(n flN). By prop.III.1.4, if xn f Bn' 00
00
is compact and absolutely convex. Indeed, to see that E £k Ak xk into £( 1 )- £( 2 ) + i£(3)- i£(4)(£(j):=..o' k k k k k k-"' If ek - o, E Ilk xk and Ilk xk converge in EK for all lln f [o,en An]' hence the conclusion. is convergent, decompose £
PROPOSITION IV.1.10. A completing web~ whose elements are absolutely convex and closed (or even fast sequentially closed) is strict. Indeed, by prop. IV.1.9, to each strand e (k~lN) of~ corvk responds an associated sequence An (n ~lN) such that the series 00
E lln xn is fast convergent for all lln f n=1 00 If moreover
E n=1
An~
[O,An] and xn f
e
vn
1, we have
00
E
n=n+1
,
lfn
0
hence the result. 2. A TOPOLOGICAL INTERPRETATION The topological meaning of the "completing" condition has been discovered by M. Nakamura [76]. It also appears, for more particular webs, in [89], [90] and [81]. Although not essential for further developments, it should be welcomed by experts, reluctant to use the above naive definition. It moreover clearly exhibits the link between our results and Banach's results
52
[6].
about complete metric groups
Since a web is completing if and only if each of its strands is completing, we just look for the topological meaning of a completing sequence. [M. Nakamura, 75]. ~ E be a lec.s. and
PROPOSITION IV.2.1.
Bn (n E-lN) a decreasing sequence of subsets of E. The following are equivalent : a) the sequence B (n E-lN) is completing, n b) there exist a complete metric group G and a continuous homomorphism
~
s£
G into E such
~
b
that, for each neighborhood U
~
0
there exist n E- lN ~ A ) 0 such that
(the unit element) of G,
a
Let Un(n E- lN) be a base of neighborhoods of 0 in G such that Un+ 1 + Un+ 1 C Un for each n E- lN. There exist such that ~ Bk n
c
~(Un)'
= 0
~(y
E- Bk and ~k E- [O,Ak], for each n, ~k n
~
~
n=1 ous homomorphism, ~
f ~
if k ~kn for each n. If
n
Yn E- Un. Thus
0 and kn
~~ E- [o,~n] =~nand Ak
Define then Ak
>
~n
yn converges in G and, since
~
n
) with
is a continu-
~
~
n=1
~k
converges in E. So does
xk n
n
~ ~k xk' which differs from the preceding one by adding vak=1 nishing terms.
a
=>
b ~
Choose Ai
) 0 ( i E-lN) and n.~ f
~
such that
~i x 1 converi=1 ges for all ~i E- [O,Ai] and xi E- Bn. • We may assume that ~
Ai !
o. Put
53
B!
l.
or
course Bj_+1 n Ak = { E i=1
c
B! for each i E- IN • Define next, for k l. l.
1'
n
co ( i
X.
~
E
i=1
B' • i2k
It is clear that
Moreover, Ak+ 1 + Ak+ 1 C Ak' ~k ~ 1 • Indeed, for each n, n
n
n
n
C E B1 + E B' E B1 + E B1 C A 2i2k k. i2k+ 1 i=1 i=1 i2k+ 1 i=1 (2i-1 )2k i=1 By prop. IV.1.7, for every neighborhood U of zero in E, An C U for n large enough. There is a unique topology t
on E which makes E into a addi-
tive topological group and admits Ak(k
e-m)
as a base of neigh-
borhoods of zero. Moreover, t is finer than the initial topology t
0
of E, Et is metrizable and, for each k, B'k C Ak. 2
Let G be the completion of E with respect to t
it is a me-
tric complete group. We are left to show that the identical map i of Et into E extends to a continuous homomorphism into E. co
~
of G
Given x E- G, there exist x 0 E- E and xn E- An such that co
E xn converges to x in G. The series E xn is also convergent n=o n=o in E for t 0 • Indeed, each xn (n~1) has the form kn xn
'"'
(n)
'"' y ' i=1 i2n
where y(n) E- B' for each i. It is possible to associate to i2n i2n each (i,k) (i E-IN, k~1) an integer 1 = y(i,k) such that y(i,k)
54
zk ~ B~ for each k, the series
E k=1
zk converges to z in E. On
the other hand, for each i, N E
N
E
k=1
zk -
~
n=1
U (B!+B! 1 + ••• +B 1 n)i 1 1+ n
c.1
)
for N large enough. For every neighborhood U of zero in E, by prop. IV.1.7, we have Cn C U for n large enough, hence N E n=o for
X
n =
X
0
N E xn n=1
+
the topology t
X
+
0
Z
Z
X
when N ""• It is obvious that z depends only on x and not on the choice 0
X
of the x~s. Indeed, if x = Exn = Ex~, then E(xn-x~) hence for t •
0 for t,
0
Defining ~(x) = z X , we obtain a extension of ~ to G, which is still a continuous homomorphism. Hence the conclusion. REMARK. When the sequence B (n n
fm)
is strictly completing, the
above proof can be simplified. Let A.k(k ~m) be a strictly associated sequence. By taking a subsequence of Bn(n ~lN), we may assume that A.k
I
0 for each k ~
m.
Moreover, it is not a res-
triction to assume that A.k+ 1 ~ A.k/2 and E A.i Bi C A.k Bk, Vk f i=k+1
m •
It is easy to check that A.k Bk(k flN) is a base of neighborhoods of zero of E for a topology t which makes E into an additive metric group. Moreover, t is finer !han t
0 •
Now Et is complete.
Indeed, if xk f A.k Bk for each k, E xk converges to some x k=1 in (E,t 0 ). Moreover,
N x -
oo
E xk = E xk f k=1 k=N+1
A.N BN' VN •
Thus x is the limit of Exk for the topology t. However, for the converse, the condition
55
co ~
)..k :Sk C :Sk '
k=k +1
lfko E- lN '
0
0
which plays a fundamental role in the theory of strict webs, does not seem to have a nice interpretation in the group topology.
3. SOME EXAMPLES PROPOSITION IV.3.1. Every
~-space
has a strict web.
Let Uk(k E-lN) be a base of absolutely convex and closed neighborhoods of zero in E such that 2Uk+ 1 C Uk for each k E-m. Then (n1, ••• ,nk)- n1 u1 n ••• n nk uk is a strict web
~.
It is first a web, generated by the sets
muk (m,kE-lN) as in example IV.1.2. Moreover, for each strand e(
n1 , ••• , nk.
) (nk E-lN,k. T co), A. = 1/sup nk is a strictly asso1 1 lc::!::k.
1
1
ciated sequence. REMARK. If E is a :Banach space with closed unit ball ( n1 ' • • • 'nk) -
:s,
n1 :S
is still a strict web and any sequence )..i (i E-lN) such that
co ~
i=1
)...
~
1 is associated to any strand of this web.
1
PROPOSITION IV.3.2. The dual E 1 of a metrizable space E, equipped with any locally convex topology for which the eguicontinuous sets are bounded, has a strict web. REMARK. A subset of Eb is bounded if and only if it is equicontinuous. Thus the finest lccally convex topology for which the equicontinuous sets are bounded is the bornological topology associated to Eb. It is also obviously equal to the topology of the indue ti ve limit of the Euo ( U f U), where U is a base of neighborhoods of zero in E.
56
The important fact here is that prop. IV.3.2 is true for all the usual topologies
t~
of E 1
where~
,
C
~b.
Proof of prop.IV.3.2. Let Un (n E-lN) be a base of neighborhoods of zero in E. We have E 1 The web
~
=
U n-=1
uo. n
of E 1 defined by
is a strict web. Indeed, the sequence 2-k(k E-E) is associated to each strand : if xk E- U~
1
for each k
e-m,
E2-kxk converges
to some point in U0 (since the e( ) are absolutely n1 n1 '• • • 'nk convex, there is no use to consider ~k E- [0,2-k]). Comment. At this stage, it already appears that the "size" of the elements of a web can widely vary from a space to another in one case, they are neighborhoods of zero, in the other one, they are compact for a(E 1 ,E). The webs encountered in the previous two examples don't use the full power of the definition : regardless to the great number of indices, the e(
n1 '• • • 'nk
)'s don't differ very much from
one another. The following example is fundamental since it includes the space of distributions of L. Schwartz. It also uses the full power of the definition. PROPOSITION IV.3.3. lL E is the inductive limit of a sequence of metrizable spaces En (n E-lN), its dual, equipped with any lecally convex topology for which the eguicontinuous sets are bounded, has a strict web. For each n E- lN, let
U~n)(k E-lN)
be a base of neighborhoods
of zero of En• For each n,
~ U~n)o = k=1
E1
,
thus
(k)o ) U••• U U ( k, n 1 , ••• , nk E- lN nk is a web. Given a strand, if xk E- e(n tions of the xk's to E.
~
1 ' ••• '
n ) for each k,
the restric-
k
( i)
belong to U ni
0
for k
~
i. Thus
xk(k E-JN) is equicontinuous on each Ei, hence on E. Thus E2
-k
00
E
2
xk converges in E• and -k
\lk
xk E-
k=k +1
0
•
0
4. PERMANENCE PROPERTIES The existence of a completing or of a strict web in a space E has a very good stability with respect to the standard operations on 1. c. s. Consider first subspaces. PROPOSITION IV.4.1.
~
(resp. strict) web in E.
E be a l.c.s.
!h£li, 1£
and~
a completing
L is a fast sequentially clo-
sed vector subspace of E, ~· = {enL
+¢ :
e E- ~~
is a comple-
ting (resp. strict) web of L. (k e-:m) of~, by prop.IV.1.9, we can choose "k the associated sequences >..k(k E-lN) such that the series For each strand e
00
E k=1
Ilk xk (Ilk E- [o,xk], xk E- e
) "k
are fast convergent. I f xk E- L for all k E- lN' they also c onverge in L. Thus >..k(k E-lN) is associated to the strand e n L(k E-lN) "k of the web ~·. We examine next the image by a linear map. PROPOSITION IV.4.2. ~ E be a l.c.s. and~ a completing (resp. strict) web in E.
1£
T is a linear map of E onto a l.c.s. F,
mapping fast convergent sequences of E into bounded subsets of F,
58
.1.!:!JUl T.1t = {Te : e
E Sll}
is a completing {resp. strict) web of F.
It is clear that TS'l is a web of F. Let e
vk
(k ElN) be a strand co
in .9l and .).k(k EJN) an associated sequence such that (Ilk E [O,.).k]' xk E e
vk
I: Ilk xk k=1 ) is fast convergent in E. By prop. III.
I: Ilk Txk is fast convergent in F, k=1 sociated to the strand Te (k EIN) of TS'l. vk If .9l is strict and the .).k 1 s such that I: k=k +1
1.12,
is as-
Ilk xk E
0
we have I:
k=k +1 0
Ilk Txk = T( I: k=k +1 0
Ilk xk) E Te
vk
0
hence TS'l is also strict. COROLLARY IV.4.}. A completing (resp. strict) web of a l.c.s. E is still a completing {reap. strict) web for Et' where t is any locally convex topology coarser than the associated ultrabornological topology. Indeed, bounded Banach disks of E are bounded in Eub' hence in Et. Apply then prop. IV.4.2 to the identical map of E into Et. Using example TII.2.4 and cor.IV.4.3, it would be enough to prove prop.IV.3.2 and prop. IV.3.3 only for the topology a(E•,E~ COROLLARY IV.4.4. If a l•c.s. E is webbed (reap. strictly web~),
so is E/L for each closed vector subspace L ~E. Indeed, the quotient map is continuous.
REMARK. We have seen in prop.IV.1.10 that a completing web .9l is strict if its elements are absolutely convex and closed. It may seem reasonable for pratical purposes to restrict somewhat the generality by considering only such particular webs. It is however too restrictive. A first reason is that prop.IV.4.2 would fail, unless very special assumptions on T are stated, since continuous images of closed sets are usually not closed. The
59
next result provides important examples of strict webs which elements are not closed. If a 1 • c • s • E ,o~:i;.::s:.,._t.::,h~e~u~n=i~o:,:;n~~o;:;,f_...::c~o:::.u=no.lit..:iia~b:::.:.lo~.Y....::m:::a~n~y
PR OP OS IT I ON IV • 4 • 5 •
subspaces Ei (i ~lN) such that each Ei is webbed (resp. strictly webbed), ~ E is webbed (resp. strictly webbed). I f 9i ( i) =
{e ( i )
v of each Ei, then
e
n1
- E -
:
~
v
I}
is a c omp 1 e t i ng (res p. strict) web
• e( ) n1 ' n1 '• • • 'nk
is a web in E.
( n1) e(
~' • • • 'nk
)(k,n 1 , ••• ,nk ~lN)
It is completing (reap. strict) since, excepting
their first element,
its strands are strands of some 9i(i)•
COROLLARY IV.4.6. A countable inductive limit of webbed (resp. strictlv webbed) sp~ces is webbed (reap. strictly webbed). Let E be the inductive limit nf the webbed spaces E.(i ~~). l.
If t i
~
is the topology of E and ti the topology of Ei, for each lN,
t
is coarser than ti in Ei. Thus Ei is webbed for the
topology t
(cor.IV.4.}).
The conclusion follows from prop.IV.4.5. Same argument for strict webs. For instance countable inductive limits of strict webs. However, except for
£~-spaces,
~-spaces
have
the elements of the
webs are generally not closed. It does not make any difference if, instead of taking vector subspaces Ei of E, we consider the more general situation of an inductive system (E. ,j. ). Simply, use prop.IV.4.2 instead of l.
l.
c or. IV. 4. } • PROPOSITION IVo4o7• .!:&1 Ei(i ~lN) be a seauence of webbed (reap. strictly webbed) l.c.s. and jik(i
s£
60
Ei ~ Ek such that
Then the projective limit E of the spaces E. (i flN) is webbed l.
{resp. strictky webbed). Let gi be the canonical map of E into Ei and, for each i flN,
m{i)
~
fe
{') 1
:
v
v f L} be a completing {resp. strict) web of E l..•
A web fR. can be defined on E as follows -1
e(1) n1
e n =g 1 1
n
e ( n1 'n2)
-1
-1 g2
(1)
e n ,._n 1 1)( 1 ")=g1 1 2 ,n1 , n~,n 2 ,n 1
and so on, where (n 2 ,n1), an integer.
'
{1)
-1
en 1 ,(n 2 ,n;)=g1
1
{2) en1
n-1
{*)
'
{2) n-1 e(n.;,n2) g~
e(n 1 ,n 2 ,n~) g2
(n~,n~,n~), ••• should be relabelled by
It is trivial that fR. is a web. An arbitrary strand e
vk
(k E-lN) of fR. is obtained as follows
:
given sequences nk(k fni), nk(k fm), ••• , the strand is a subsequence of -1 e ( 1 ) g1 n, -1 -1 (2) eI ~1) g1 en' 2 e n1 'n2) n g2 1 -1 -1 (2) el = g1 ~1) e ( n 1I ,n 2I) n e n 1 ,n 2 ,n~) n g2 ~ el
1
.••
.••
-1 e(~) n"1
g~
••
••
•
•
( c 1)
(c~)
( c2)
•••
There comes out of the columns (c.) a strand of each fR.(i).
>..~1)(kflN)
be associated to the
f~rst
column,
>..~ 2 )(kE-lN)
Let
to the
second, and so on. Assume for a while that none of them is vanishing. Then
(1)
11 1 = >.. 1
(1)
, 11 2 .. inf ( >.. 2
,
(2)
>.. 1
.
(1)
, 11 ~ = J.nf ( >..~
is associated to the strand of fR.. Indeed, if xk f
e
(2) (~\. , >.. 2 , A.1 '), • ••
vk
, for k
~
i,
g 1 (xk) belongs to the strand corresponding to the ith column and, since the 11 ks are less than or equal to the corresponding asso-
61
00
ciated numbers,
~~ gi(xk)
E
converges in Ei for all
k=1 DO
~~ ~
[o,~k].
E
Thus
~k xk converges in the projective limit
E.
k=1
Now, if the
1~ 1 ) .t{
are not
all strictly positive, take a sub-
sequence of non vanishing terms. The selected indices fix a subsequence of the second strand, thus a new strand in E 2 • Take t..~ 2 )(k ~lN) associated to it and repeat the argument. The end of the proof is then the same as above. RENARK.
It is essential in this proof that a subsequence of a
strand be another strand.
Our definition of completing webs in
[14] assumes that each strand of the form e(
n 1 , ••• , nk
)(k ~JN)
has an associated sequence of strictly positive numbers.
In or-
der to get the topological interpretation of the webs of
§2,
it
is necessary to allow these numbers to cancel. To preserve prop. IV.4.7, we are then obliged to take strands of the form e
n 1 , ••• ,nk.
(i ~lN), with k 1 t
DO•
l.
PROPOSITION IV.4.8. Every countable product of webbed (resp. strictly webbed) spaces is webbed (resp. strictly webbed). Let Ei(i E-lN) be l.c:s. with completing or strict webs .9l(i)• DO
n
The product
Ei is the canonical projective limit of the fi-
i=1 N
nite products
n
E .• Thus, by prop. IV.4.7, it suffices to coni= 1 l. sider finite prodncts. By induct:Lon, it is even sufficient to
deal with the product of two spaces. The same proof would work for the general case but it is much simpler to write it down for two spaces. Define on E 1 x E 2 the web X
(2) e(n, '• • • ,nk) '
indexing each (mk,nk) by a single index.
62
A strand is obtained by fixing sequences mk' nk (k E-lN) and k 1 f ~ 1 denote it ei
=
e( 1 ) x vi
e(~) vi
with
e( 1 ) .. e( 1 ) vi
(m,, ••• ,mki)
; e( 2 ) = e( 2 )
vi
(n,, ••• ,nki
)(ie-m) •
Let A,.(iE-JN) be a sequence associated to e( 1 )(iE-lN). Let 1 vi iJ.(j E-lN) be a subsequence such that A. • .:/= 0 for all j . Then 1j
e(~) v.
is a strand in 9l( 2 ). I f IJ.J.(j E-JN) is associated to it, the
1j
sequence £i (i E-lN) where £i
0 if A.i
=
0 and
£. 1.
=
inf
J
is associated to ei(i E-lN). PROPOSITION IV.4.9. Every countable direct sum of webbed (resp. strictly webbed) spaces is webbed (resp. strictly webbed). It is the inductive limit of the finite products of the spaces. Hence the conclusion, by prop. IV.4.8 and cor.IV.4.6. 5. CLOSED GRAPH THEOREMS Let us start with the simplest case, in order to clear the proof of all cumbersome technicalities. PROPOSITION IV.5.1.
~
E,F be locoso and T be a linear map of
E into F. Assume that E is an ~
~-space
, F admits a strict web
G(T) is fast sequentially closed in E x F. ~ T is con-
tinuous. Let Uk (k E-lN) be a base of neighborhoods of zero in E such that 2Uk+ 1 C Uk for each k E-lN and let 9l = {ev 1 v E- I} be a strict web of F. We essentially duplicate the proof of prop. I.2.1.
Step a. Since
= T- 1F
E
T- 1 e
u n 1 f IN
n1
by prop. 1.1.2, there exists n 1 f
m
such that T- 1
is non
meagre in E. From U
T
-1
n 2 f lN
e(
n 1 ' n2
, , 1
again by prop.I.1.2, there exists n 2 such that T
-1
e(
non meagre.
n1 'n2
) is
Hence, by induction, there exists a strand e
(n E-lN) of the vn (n E-lN) is non meagre for each n E-lN.
web fJl such that T- 1 e
vn By prop. 1.1.3, since the T- 1 e
T
-1
vn
are absolutely convex, the
are neighborhoods of zero in E : there exist k n vn -1 , Vn f IN. such that uk C T e vn n
e
T
~
Step b. Let >..n(n E-lN) be strictly associated to the strand e (nE-lN). We may assume, by considering a subsequence, that vn An =I= 0 for all n. Using prop. IV.1.8, we can choose An (n E-lN) ~
E An yn is fast convergent for all yn E- e • Finally, vn n=1 changing kn (n E-lN), we may suppose that
such
uk
-1
c
An T
c
T-1 e
c
-1 An T e
n
e
'
vn
vn f lN.
Then uk
vn-1
n
vn > 1 •
'
Indeed, uk
n
Given xn 0 f
Uk
and Yn f T- 1 e
64
, n0
vn
c
vn
A T- 1 e + uk n vn n+1
'
Vn •
there exist then sequences xn E- Uk (n>n 0 n
(n~n ) such that o
)
Summing up these relations for n
~
0
n
~
N, we get
N I:
n=n
0
hence
On the other hand, by definition of strictly associated N I:
sequences,
A n Tyn is fast convergent in F to some point
n=n
z
~
e
vn -1 0
.
thus (xn 'z)
0
Since G(T) is fast sequentially ~
G ( T), hence
X
0
n
~
T-1
0
e
vn -1
C
1 OS ed,
and uk
0
we have
< T-1 n
e
0
v n -1 0
SteD c. By prop. IV.1.7, for every neighborhood V of zero in F, A 1e M
~1
C V for n large enough. Hence T(An+ 1 Uk
n
) C V and T is
continuous. Comment. A first extension of prop. IV.5.1 follows from prop. II.3.12 : we can replace the
~-space
E by an ultrabornological
space. The three other natural extensions consist in 1) considering linear relations instead of linear maps, 2) dealing with more general webs, 3) as in prop. I.2.2, replacing E by a Baire space (the repeated use of prop. I.1.2 prevents from taking E barrelled). The next result can be regarded as the general closed graph theorem for webbed spaces. It is the main result of this section. PROPOSITION IV.5.2. ~ E ~ F such
~
E,F be l.c.s. and R a linear relation
that ~(R)
=
E. Assume that E is ultrabornolo-
sical, F webbed and G(R) fast sequentially closed in E x F. Then R is continuous.
65
In particular, i f R is a linear map of E tinuous and if RE is an
~
~ v(R)
1
~
is a linear map of F onto E, R
~-space,
F, it is con-1
is open,
is still true if we assume
~result
is non meagre in E and Yields V(R) = E,
By prop. II.3.12, we may assume that E is an ~-space (and even a Banach space). Let Uk(kE-JN) be a base of neighborhoods of zero in E and ~
!e
v
As in step a of prop.
: v E- I} be a completing web in F, IV.5.1, there exists a strand -1
(n E-lN) of ~ such that R e is non meagre for each n E- JN. vn vn Observe that this requires only R- 1 (F) = V(R) not to be meagre
e
to start the induction process, rather than V(R) Each R
-1
=
E.
e
has thus an interior point (but it is no lonvn ger a neighborhood of zero). Let An(nfJN) be associated to the strand e a subsequence, we may assume that An Choose xn and kn f A
R
n
-1
e
~
such that
vn
We may assume th a t xn ~~ ~'n R-1 e 1
( xn + 2u k
n
)
x~ +
t i on,
vn
: indeed,
; for every x'n in this intersec-
n An 1
2
+
(n fJN). Taking vn 0 for each n.
C A
Uk
n
n
R
-1
e
vn
Let V be an absolutely convex neighborhood of zero in F. By prop. IV, 1 • 7, co
I
E
i=n
( *)
Ai yi :
for n large neough, say n
~
n
0
•
Let us prove that 2R- 1 V ) Uk
• Since n
0
'In
66
f lN
,
we determine by induction sequences yn E- Uk
given yn
n
0
(n~n ) such that (n>n ) and xln E- A. n R- 1 e 0 vn o X 1
-
n
X
+ y
n
n+ 1
t
';In ~ n
o
Thus N L: n=n
in E.
(X I -X
n
n
)
-o
y
0
n
0
On the other hand,
that
there exist z
1 Rz n and xn1 Rz n" Moreover
N L:
n=n
n
,
z
1
n
e
E- A
n
v
such n
(z 1 -z ) is fast convern
n
0
(*).
gent in F and its limit z belongs to 2V, by
Thus yn
Rz 0
(because G(R) is fast sequentially closed) and y
n
E- 2R-1v. 0
PROPOSIT:ON IV.5.). Prop.IV.5.2 is still true when E is an inductive limit of Baire spaces, i£ G(R) is closed instead of fast seguentially closed. It is also true with no assumption on E if ~(R) is non meagre in E and it yields ~(R)
= E.
The latter condition is an improvment of "~(R)
E" but not
of the assumption on E since, if ~(R) is non meagre in E, E is non meagre in itself, hence a Baire space. The proof has to be modified as in prop. I.2.2. Repeating the above arguments, we obtain a strand e
(n E-lN) vn of fll and a sequence of neighborhoods Un(n E-lN) of zero in E sud that
un
-
X
n
• However Un(n E-lN) is no longer a base of vn neighborhoods of zero in E. e
there exist
Following the preceding proof, given yn 0
x 1 E- An R- 1 e (n~n o ) such that n v n
67
Yn = x~- xn + Yn+1' vn ~no • Let U be an arbitrary neighborhood of zero in E. We have N E
n=n 0 Since
we Hence N
E (x~-xn) - yN+ 1 + xN+ 1 f n=n 0
u.
On the other hand, if yN+ 1 R wN+ 1 , with wN+ 1
~
XN+ 1
N E
n=n 0 is still convergent in F and its limit z belongs to 2V. Thus, since G(R) is closed, we have again (yn ,z) f G(R) and Yn
f
2R- 1 V. Hence the proof.
0
0
Let us come back to prop. IV.5.2. We have seen in prop. IV.5.3 that the assumption on E can be replaced by an assumtion on ~(R), but this is not a real progress on E since anyway it must be a Baire space. Can we now relax the assumption on F? A first obvious answer is that it is enough to assume that R(E) is webbed, instead of F, since it makes no difference to replace F by R(E) • When E is an U-space and F is webbed, E x F is webbed (prop. IV.4.8) hence, if G(R) is fast sequentially closed in E x F, it is also webbed (prop.IV.4.1). So it would be an improvment to take G(R) webbed instead of assuming that G(R) fast sequentially closed and F or R(E) webbed. The eventual progress is however not significant with respect to F since, if G(R) is webbed, R(E) is webbed because it i the image of G(R) by the canonical projection of E x F onto F
68
(prop. IV.4.2). It is however an improvment on G(R), since it can be webbed without being closed nor fast sequentially closed. PROPOSITION IV.5.4. ~ E ~ F.
1L
~(R)
~
E,F be l.c.s. and R a linear relation
is non meagre in E ~ G(R) webbed~ R ~
continuous. of E into G(R) by xR'(x,y) ~ ) xRy. It is still linear and ~(R) = ~(R•). Moreover, R•- 1 is simply Define a new relation R'
the canonical projection and G(R
1 )
n1
of G(R) in E. Thus it is continuous
is closed. By prop.IV.5.3, R' is thus continuous.
This implies the continuity of R. Indeed, R
~
n 2 o R 1 , where
n 2 is the projection of E x F onto F.
6. LOCALIZATION PROPERTIES A simple look at the proof of prop. IV.5.1 shows that, when E is an v-space, F a strictly webbed space and T a linear map of E into F with a fast sequentially closed graph, we get inclusions of the type "TU C e" where U is a neighborhood of zero in E and e an element of the web of F. This has been shown to imply the continuity of To But, since the elements of the web may be much smaller than neighborhoods of zero, it is worth to see what extra information this brings on T. PROPOSITION IV.6.1. tion of E
~
~
E,F be l.c.s. and R be a linear rela~(R)
F such that
= E. Assume that E is an V-spa-
that F has a strict web ~ = {e : v f I} and that G(R) ~ v fast sequentially closed in E x F.
~'
Then, i f -----
v
>
R
-1
e
~
is a neighborhood of zero in E.J there exists -1
-
~ such that R e is also a neighborhood of zero in E. v In particularJ there exists a strand e (kfm) such that all vk -1 R e are neighborhoods of zero and we can choose for the first vk -1 element any v such that R ev is a neighborhood of zero.
69
Assume that R
-1
e
~
is a neighborhood of zero in E. It is thus
non meagre. Since R -1 e that R
-1
u R-1 e 'II
~
,
v>~
there exists v 1
is non meagre too. v1 By induction, we can thus find a strand e
R
-1
>
such
~
e
vn
(n E-lN) such that
e
+¢ for each n. Choose then a strictly associated sequenvn ce An (n E-lN) and, assumin,g that >..n =I= 0 for each n, a base of
neighborhoods of zero U (n E- JN) in E such that Un n for each n.
<
An R
-1
e
vn
An argument similar to step b in prop. IV.5.1 shows that Un
<
R- 1 e
'~~n-1
for each n E-lN, hence the proof.
For the second assertion, it is enough to find ~ v such that R- 1 e is a neighborhood of zero to start the procesG. But R -1 e¢
=
v -1 R (F)
=
E, hence the result.
PROPOSITION IV.6.2. Prop. IV.6.1 is still true if ~(R) is non meagre in E (~ E is a Baire space) ~ G(R) closed in E xF. Modify the above proof as the one of prop.IV.5.3· A similar property exists for completing webs. PROPOSITION IV.6.3. Assume that E is an ce with completing web
~
=
{e
=
v
~-space,
F a webbed spa-
: v E- I} ~ R a linear relation
in12 F such that ~(R) closed in E x F. If R- 1 e is non meagre in E, then R- 1 Co(e ) is a neighbor-
~ E
-
~
E and G(R) is fast sequentially
-
~
>~
hood of zero in E and there exists v
such that R- 1 e
v
is non
meagre in E. In particular, there exists a strand e R -1 e
are non meagre and R -1 Co( e vk
)
"k
vk
(k E-lN) .:;:S..:ou~c~h~~t:.:h.lil:a..:t:.....:a::.l~l
neighborhoods of zero. :tfJ!
can moreover choose for the first element, any e such that v R- 1 e v l.S · non meagre.
70
The result is still true without assumption on E
ii
~(R)
is non meagre in E ~ G(R) closed in E x F. The proof is entirely similar. First fix a strand e such that all R
-1
e
vk
xn E- An R
e
vk
E- Un , 0
IV.5.2,
and neighborhoods of zero Un such that
According to the proof of prop. Yn
(k E-lN)
are non meagre. Take then an associated
sequence An(n E-lN) and determine as in prop. -1
vk
rv.5.3, we get that,
for every
there exists
0 00
z
=
( z n1 - z n ) ( z n , z n1 E- An
L: n=n
0
If we assume that An+ 1 ~ An/2 for each n,
Rz.
such that yn
it
0
2 ~(e
is clear that z f
), hence the conclusion. vn -1 0
We have in fact proved a sharper result which will be useful later. PROPOSITION IV.6.4.
Under the assumptions of prop. IV.6.3,
there exists a strand e
vk
(k flN) such that, for each associated
sequence Ak(k fiN), 1
00
00
R- ( L:
Ak e
k=1
vk
is a neighborhood of zero in E. COROLLARY IV.6.5.
~
E be an
~-space,
F the inductive limit
of a sequence of webbed spaces Fi (i E-lN) ~ T a linear map of E into F. If the graph of T is fast sequentially closed, then there exists i E-lN such that TE into F .•
- l .
<
F.
l.
It is still true if G(T)
c 1 os ed in E x F i
.;f...:o~r~..::e~a:!;;c=h
and that T is continuous of E
n
(ExE.) is fact sequentially l.
i E-lN.
Assume first that the spaces Fi have strict webs
~(i).
Then,
71
by cor.IV.4.6, F has a strict web ~ in which ei = Fi for each i ~JN. By prop. IV.6.1, there exists a strand e (k ~lN) such vk
that each T- 1 e
vk
is a neighborhood of zero in E. In particular,
if v 1 ~ {i 1 }, T- 1 Fi 1 is absorbent in E, hence TE C F.
1.1
continuous of E into F.
11
Now Tis
by the closed graph theorem.
If the spaces Fi have only completing webs ~i' apply prop. IV.6.4. Each strand e (k ~lN) of the web of F is a strand of vk
some ~i provided v 1 ~ {i}. Thus R- 1 Fi is a neighborhood of zero in E and TE C Fi. The map T is then continuous of E into Fi by the closed graph theorem. Let us now improve the assumption on G(T) : it does not need to be closed for all fast convergent sequences, but for those occuring in the proof, which are constructed by means of strands of the web and are actually fast convergent in E x Fi for some i
~IN •
This is an example where G(T) is not a priori closed but still webbed (cf.prop.IV.5.4). Corollary IV.6.5 has been proved for strict ~-spaces by Dieudonne and Schwartz [30] and for ~-spaces by Kothe. COROLLARY IV.6.6. If a Baire space E has a completing web, ~ is an fF-space. Let E be a Baire space and ~ a completing web of E. By prop. IV.6.3 applied to the identical map of E into itself, there exists a strand e (k ~IN) of ~ such that "Cc;(e ) (k ~lN) are neighvk
vk
borhoods of zero in E. For suitable A.k(k ~lN), A.k ~(e
vk
)(k ~lN) is a base of nei,e:h-
borhoods of zero in E (prop.IV.1.7). Thus E is metrizable. Denote ~y i its completion. The space E is a non meagre subset of
i.
00
Indeed, assume that E C
U Fn' where the F's are n=1 n closed in i. Since E is a Baire space, some Fn n E has an interior point in E Fn ) x + U, where U is a neighborhood of zero in E. Then Fn ) x + U (closure in E) and U is a neighborhood of
72
A
zero in E. Consider again the identical map I is non meagre in
E,
of
E into E. Its domain E
its graph is closed in
Ex
E and E is web-
bed. Thus, by prop.IV.5.2, the domain of I is equal to
E.
Hence
the result. We conclude this section with a slight improvment of the assumption on the domain space, when the range space has a special kind of web. DEFINITION IV.6.7.
Let E be a l.c.s.
We say that B C E is
~-bounded
of ~ and a sequence >..k
>
and~
a strict web of E.
if there exists a strand e
vk
( k f JN)
0 (k fiN) such that
00
B C
LEMMA IV.6.8. 1,tl ~
E is
~be
~-bounded
a strict web of E, Every Banach disk B
and every
~-bounded
subset of E is bounded.
The first part follows from prop.IV.6.1 applied to the identical map of EB into E. The second one follows from prop.IV.1.7. web~
DEFINITION IV.6.9. A strict ~-bounded
is called reflexive if every
subset of E is relatively weakly compact.
LEMMA IV.6.10.
l! E is semi-reflexive, every strict
web~~
E
is reflexive. The converse is true if every bounded subset of E ~~-bounded
(~,
for instance, if E is quasi-complete).
Straightforward by the above lemma. DEFINITION IV.6.11. Let
~bd
be the collection of all bounded
Banach disks of E and Ebd be the space E 1 equipped with the corresponding topology
•
t~
bd We have seen in prop.III.2.9 that E is ultrabornological if and only if it is a Mackey space and E£c is complete. It is well known that, if E£c is complete, Ebd is complete too. The conver-
73
se is however false in general. Indeed, let E = F', where F is an
~-space.
'f
We have E' =F.
Moreover, every bounded Banach disk of E is equicontinuous on F and, conversely, every equicontinuous subset of F' is contained in a bounded Banach disk of E. Hence Ebd is F equipped with its natural topology and it is complete. On the other hand, if Efc is complete, since E is a Mackey space, by prop.III.2.9, E is ultrabornological. Then at least E = F'
'f
(cf.ex.III.2.5) is
also equal to Fb and F is semi-reflexive. The next result finds its origin in [70]. PROPOSITION IV.6.12. ~ G(T) ~
~
T be a linear map of E into F,
~
is closed in E x F. Assume that E is a Mackey space,
Ebd is complete and that F has a reflexive web.
~
T is
continuous. First, for every bounded Banach disk B of E, T maps B into a relatively weakly compact subset of F : apply prop.IV.6.1
to T
restricted to EB. Since TB is moreover absolutely convex, it is contained in an element of $
'f
(F).
By the closed graph theorem (prop.IV.5.2), T is continuous of Eub into F. Since it maps $bd(E) into$ 'f (F), T* is continuous of F' into (E b)' equipped with the topology t$ (E)" 'f u bd -1 The subset T* E' is actually ~(T*) (T* regarded as a map of F 1 into E 1
).
Since Ebd is complete, it is closed in (Eub)' for the
topology t$bd(E)' hence ~(T*) is closed in F'. 'f But, by prop.I.2.5, it is dense in F' for a(F 1 ,F) hence for the Mackey topology
'f•
Thus ~(T*)
=
'f
F1
•
The conclusion follows by prop.I.2.4. COROLLARY IV.6.13.
~
E be evaluable and complete and let F
11.
admit a reflexive web :Jl. ~
T
~
74
'f
G(T) is closed in E' x F, ~ T is continuous and there
exists a linear map T 1 1
T is a linear map of E' .!..!!ll F such
'f
~
F1
1n12 E such that T 1 *
=
T. Moreover
is continuous for the topology t:Jl of uniform convergence on :Jl-bounded subsets of F and the natural topology of E.
The space E' is a Mackey space. Moreover, since E is evalua't
ble, the bounded Banach disks of
E~,
being bounded in Eb, are
equicontinuous. Thus (E~)bd is E equipped with its natural topolcgy. The assumptions of prop.IV.6.12 are thus fulfilled and T is continuous of E 1 into F. 't
Its adjoint T* is thus defined of F• into (E')' =E. Since 't
maps the equicontinuous subsets of E'
into
~-bounded
't
subsets of
F, by duality, T* is continuous of Ft into E. ~
For all x' Thus (T*)*
~
E 1 andy'
~
F1
,
we have
1)
.
T.
There are interesting examples of reflexive webs, even in non semi-reflexive spaces. For instance, we have seen that the dual E 1 of a metrizable space or of a countable inductive limit of metrizable spaces is strictly webbed (prop.IV.3.2-3). Moreover, its web
~
is such that every
~-bounded
subset is equicontinuous.
It is thus relatively weakly compact when E 1 is equipped with the Mackey topology (or any coarser Hausdorff topology). case, the topology t~ defined above on (Ei)'
=
In this
E is simply the
natural topology of E (every ~-bounded set is equicontinuous an~ conversely, every equicontinuous set is contained in a bounded Banach disk, hence ~-bounded by lemma IV.6.8). COROLLARY IV.6.14.
~
E be evaluable and complete and F be me-
trizable or a countable inductive limit of metrizable spaces.
l£ T is a linear map of E' into F 1 such that G(T) is closed in E 1 x F', 't
~Tis
the adjoint of a continuous linear map ofF
't
i!!..l.Q.
E•
It implies in particular that T is continuous for a lot of topologies on E 1 and F 1 (for instance of E's into F's' of El, into F and so on) • It is a straightforward consequence of cor.IV.6.13 and of the
b,
above comments on the web ofF'.
75
V Webs in spaces of linear maps
1. SPACES OF LINEAR MAPS DEFINITION V.1.1.
Let E,F be l.c.s. We denote by L(E,F) the spa-
ce of all continuous linear maps of E into F. If family of bounded subsets of E, we denote by
t~
~
is a covering
the topology de-
fined on L(E,F) by the semi-norms rB
,p
where B f
=
(T) ~
sup p(Tx) xfB
and p is a continuous semi-norm on F.
The space L(E,F) equipped with the topology t~ ,t~ , ••• is s b denoted by Ls(E,F), Lb(E,F), •••• Moreover, we denote by p the finest locally convex topology on L(E,F) for which the equicontinuous completing disks of L(E,F) are bounded. It is clear that p is finer than t~ for each
~.
The next example will help to see how to find webs in spaces of linear maps. EXAMPLE V.1.2.
1L
E is metrizable and F ~~-space, Lp(E,F) ~
a strict web. Let Uk(k flN) and Vk(k flN) be bases of neighborhoods of zero in E and F. Define e(k) n
!T f
L(E,F)
The sets ( 1)
e(n1, ••• ,ni) = e n1 determine a
76
web~
( i)
n ••• n e n. l.
of L(E,F).
Given a strand e
vi
= e(
n 1 , •• , , nk. )
(n
k
E-lN, k.
l.
T co)
of fll,
l.
for all i, we have c vk,
11i ""'k ,
thus the sequence T.(iE-lN) is equicontinuous. Since F is coml.
plete, every closed and absolutely convex equicontinuous subset of L(E,F) is completing. The conclusion follows by prop,IV.1,8 and 1 o, The localization properties (IV,§6) suggest that the elements of a web of F could play the same role as the Vk ( k E-lN) in the above example, PROPOSITION V,1,3,
1£
E i s a Banach space or a countable induc-
tive limit of Banach spaces and if F has a strict web,
then
Lp(E,F) has a strict web. Assume first
that E is a Banach space, with closed unit ball
B, and that F has a strict web fll, By prop.IV.6,1, for each T E- L(E,F), there exists v E- I and m E- lN such that TB C m e there exist v'
>
and, if this is true for v and m, v v and m' E- lN such that TB C m' e 1 • v
According to prop,IV.1.6, we can change the web fJl in such a way that we no longer need the constants m,m•. Define then
&v
{T E- L(E,F) : TB C e } (v E- I), v
We have just seen that L(E,F) = U & v
v
and &
Jl
&
v
hence fll' {& v E- I} is an absolutely convex web of L(E,F), v It is a strict web. Indeed, take a strand & (k E-lN) and a vk sequence xk(kE-JN) strictly associated to the strand e (kE-lN). vk N I f Ilk E- [o,xk] and Tk E- & { E Ilk Tk : N E- lN} is equicon\lk ' k=1 tinuous. Indeed, for every neighborhood V of zero in F, we have
77
N E
k=k 0 for k 0 large enough (prop.IV.1.7). Thus N
( E ~k Tk)B C V, VN ~ k 0 k=k 0 There exists C
>
0 such that
i ~k
E k=1
Tk B C CV, Vi ( k 0
hence N I:
~k Tk B C (1+C)V,
VN E- lN •
k=1 Moreover, if x E- B, E Tkx E- e
vk
~k
Tk x is convergent in F, since
• Hence E ~k Tk is convergent in Ls(E,F). Since
E
k=k +1
~k
0
Tk x E-
I:
k=k +1 0
~k
e
vk
C e
vk
,
vx E- B ,
0
we also have
We have thus proved that ~· is a strict web of Ls (E,F). Refering to the proof of prop.IV.1.9, for ek- o,
( *) is even fast convergent and, more precisely, convergent in the Banach space generated by the Banach disk
It is elear from the above arguments that K is moreover equicontinuous in L(E,F). Thus it is a bounded Banach disk of L~(E,F) and (*) is also convergent in L~(E,F) hence ~· is a strict web of L~ ( E, F). Assume now that E is the inductive limit of the Banach spaces Ei(i E-lN).
78
The space L (E,F) is isomorphic to the closed subspace of s
00
n
i=1 T.
l.+
L (E. ,F) made of all sequence T. ~ L(E. ,F) (i flN) such that s l. l. l. 1 restricted to E.l. equals T.1 for each i ~lN. By the permanence properties (prop.IV.4.8 and prop.IV.4.1).
Ls(E,F) has thus a strict web~·. If we refer to the construction of this web, it is easy to see that, for each strand &1
vk
(k E-lN) of ~·,
there exists an associated sequence >..k(k E-lN)
such that 00
00
{E
£kJ..kTk:
k=1
E k=1
l£kl~ 1 }
is still equicontinuous in L(E,F) for all Tk ~ 0: 1 (kE-lN). So we vk
can conclude as above that ~· is still a strict web in Lp(E,F). PROPOSITION V.1.4. lL E is an U-space or a countable inductive limit of U-spaces and F a sequentially complete and webbed space or a countable inductive limit of such spaces Fi'
~
LP(E,F) is webbed. If moreover F or each Fi has a closed and absolutely convex ~' ~ LP(E,F) is strictly webbed.
Assume first that E is an U-space and F a sequentially complete webbed space. Let U. (i ~lN) denote a base of neighborhoods of zero in E and ~ = {e
1
v
: v ~ I} a completing web of F.
If T E- L(E,F), by prop.IV.6.3, t"nere ex1.sts · · v E- I sue h that T- 1 e v l.S non meagre, if T- 1 e
v
is non meagre, there exists
~
>
v
such that T
-1
e
~
is non meagre and i E-lN such that TU.1 C ~(e v ). Define then 0:(. ) (. ) (where each (i 1 ,n 1 ), ••• , 11,n1 , ••• , 1k,nk (ik,nk) should be indexed by a single index inlN) as the set of all T E- L(E,F) such that T that
-1
e(
n1'"""'nk
) is non meagre and
79
The sets
define a web~· in L(E,F).
&(.l.1,n1 ) , ••• , (•l.k,nk )
(k ~lN) in 9i 1 • The corresponding indices vk nk(k ~lN) determine a strand in 9i. Take a decreasing sequence
Fix a strand C
Xk(k ~JN) associated to the strand e( n 1 , ••• ,nk )(k~lN) of 9i. If Tk ~ C
(k ~lN), the sequence Xk Tk (k ~lN) is equicontinuous. vk Indeed, for every neighborhood V of zero in F, by prop.IV.1.7,
for k large enough, say k
~
k , 0
Xk e( n , ••• ,nk ) C V 1
hence, if we choose V closed and absolutely convex, C
Cc;"(e( n , ••• , nk )) C V, lfk ~ k o • 1 0
Since F is sequentially complete, every closed, absolutely convex and equicontinuous subset of L (E,F) is a Banach disk. s
It is also a Banach disk in L~(E,F). Thus, by prop.IV.1.8, 9i 1 is a completing web of L~(E,F). If the elements e of 9i are closed and absolutely convex, v 1 the condition "T- e v is non meagre and TU.l. C C'O(ev)" is equivalent to 11 TU.l. C e v "• Thus
These are closed and absolutely convex subsets of Ls(E,F), thus, by prop.IV.1.10, the web 9i 1 is strict. Assume now that E is an U-space and F a countable inductive limit of sequentially complete and webbed spaces F. (i ~IN). l. By cor.IV.6.5, L(E,F) is the union of the spaces L(E,F.). l. Moreover, every equicontinuous subset of L(E,F.) is equicontil. nuous in L(E,F). Thus the topology ~ of L(E,F) is coarser in L(E,Fi) than the topology ~ of L(E,Fi). The conclusion follows then from prop.IVo4o5o Assuming finally that E is a countable inductive limit of U-spaces Ei(i~IN), we conclude as in the last part of the proof of prop.V.1.3.
80
Another proof for L (E,F) is obtained by observing that it s is the projective limit of the spaces L (E.,F) and by applying s 1 prop.IV.4o7• However the topology of L~(E,F) is finer that the topology of the projective limit of the spaces L~(Ei,F). The reason why the result is however true is hidden in a slight ~
improvment of cor.IV.4.3 : let disks of E and e
~
be a collection of Banach
be a web of E such that, for each strand
(k f I), there exists an associated sequence J..k(k fJN) such vk
that I:j.Lk ~ (IJ.kf[O,J..k], xk f e space EB for some B f
~.
vk
Then
) is convergent in the Banach ~
is still a completing web when
E is equipped with the topology of the inductive limit of the spaces EB(B f
~).
In the present case,
~
is the family of equi-
continuous Banach disks and the corresponding topology
~
could
be finer than the ultrabornological topology associated to L8 (E,F). The next properties are closely related to the above ones since the dual
E~
of a metrizable space is in a suitable sense
a countable inductive limit [41,91]. PROPOSITION V.1.5.
~
E be metrizable and F admit a strict web
(reap. be sequentially complete and webbed). Then Lb(E~,F) ~ strictly webbed (resp. webbed). Let E~ the inductive limit of the spaces E'u?' where Ui(i flN) 1
is a base of neighborhoods of zero in E. By prop.V.1.3 and 4, L~(E~,F) has a
strict (reap. completing) web when F is strictly webbed (reap. sequentially complete and webbed). The space L(E~,F) is a vector subspace of L(E~,F). Indeed, since
U~
is bounded in E 1 c in F, hence T f L(E~,F). 1
If~
,
for all T f
L(E',F), c
TU~ 1
is bounded
= {e v (v f I)} is a strict (reap. completing) web of F,
the web~· of L(E~,F) consists of finite intersections of sets & (i, v)
= {T
f L ( E, F) : TU
i
C
e) ( i
f :N, v f
I) ,
Provided the constants are ruled out by means of prop.IV.1.6, 81
or of sets {T E- L(E,F) : (T- 1 e ) n Euo is non meagre in v i and TU~ C G;(e )} l. v
Eij~ l.
•
If E' (k E-:IN) is a strand of .7! 1 and A.k(k E-lN) an associated vk sequence, for each i E-lN, there exists a strand .7t such
that
T U~l. C G;(e v (")' VT E- C'vk l. k
for k large enough. Thus, given a neighborhood V of zero in F and £ > o, for all ILk E- [O,A.k] and Tk E- C' vk ' N ( I: !LkTk)Ui_ c e:V, VN ~ k 0 ' k=k 0 for k 0 large enough (prop.IV.1.7). Therefore, if q is the gauge semi-norm of V, N
q [( E !LkTk)x 1 ] k=k 0 is uniformly convergent on cr(E 1 ,E), the limit q[(
""I:
U~. l.
Since it is continuous on
!LkTk)x 1 ]
U~ l.
for
is still continuous on each
k=k 0 Ui_ for
0
(E 1 ,E). It follows then by the Banach-Dieudonne theorem
(prop.I.2.6) that q[( "" I: !LkTk)x 1 ] "" k=k 0 I: 11kTk E- L(E 1 ,F). k=k c
is continuous onE' , so that c
0
Thus L(E~,F) is closed in L(E~,F) for the sequences considered when checking that .7! 1 is a completing web. Hence the web
!C'v
n L(E',F) : C' c v
f
.7!'}
is still completing in L(E' ,F) equipped c
with the topology induced by L~(E~,F). The topology of Ls(E&,F) is coarser than the topology of L~(E&,F) and coincides on L(E~,F) with the topology of Ls(E~,F). Thus Ls(E~,F) is webbed. Moreover, the topology of Lb(E~,F) is coarser than the ultrabornological topology associated to Ls(E~,F),
82
thus Lb(E~,F) is also webbed (cor.IV.4.)).
PROPOSITION V.1.6.
~
E be a strict inductive limit of metri-
zable spaces Ei ~ F a strictly webbed (resp. sequentially complete and webbed) space, such that, for every sequence of continuous semi-norms q.l. (i E-lN), there exists a continuous semi-
>
..!.UU:.!!l q and a sequence Xi
0 (i E-lN) such that
qi(x) ~Xi q(x), ~if lN, ~ Lb(E~,F)
~x E- F
•
is strictly webbed (resp. webbed).
Define ~ 1. of L(E! ,F) into L(E',F) by l. 'c c (~iT)x 1
=
T(x• IE.), ~x' f E•
•
l.
We shall prove that 00
L(E',F) = C
U
i=1
"t.L(E! l.
J.,C
,F) •
its restriction to E.l. is For every bounded subset B of E'c clearly bounded in E! • Thus each ~ 1. is continuous of J.,c Lb ( E ! , F ) in t o Lb(E~,F). The conclusion follows then from prop. J.,c
IV.4.5 and 2. LetT belong to L(E',F). For some i f lN, TE~l. = o. Indeed, c otherwise, there exist for each i f lN, x.l. f E!l. and a continuous semi-norm qi ofF such that qi(Txi)
I
0. There exists then a
continuous semi-norm q of F such that q(Tx.) l.
+0
and it is not a restriction to assume that q(Txi) i
(multiply x. by a suitable constant). However, l.
for each i E-lN
=
1 for each
the sequence
xi (i flN) is equicontinuous on E. It is moreover convergent to zero in E', thus, because of the equicontinuity, in E'c• Hence a s contradiction. 0. We construct T. f L(E! ,F) such that Assume that TE~ l.
l.
~iTi
J.,C
=
T. Since the inductive limit is strict, the topology induced in
Ei byE is equal to the topology of Ei. Thus Ei. = {xiE.: x E-E'}. l.
Given x f Tix
=
Y-y 1 f
El, take y f
E 1 such that YIE.
=
x and define
Ty. The definition is coherent, sine~ YIE.
Y'IE.
l.
l.
implies
Ei hence Ty - Ty' = 0.
83
It is clear that T.~ is linear. To prove that T.~ is continuous, by the Banach-Dieudonne theorem, it is enough to check that for each continuous semi-norm q on F, q(T.x) is continuous ~ for a(E!,E.) in each equicontinuous subset A of E!. If it is not ~
true,
~
there exist E:
>
~
0
and a generalized sequence xa( a E- €7) E- A
such that q(T.x ) ~ E: and x ( ~ e1)- 0 for cr(E! ,E.). By the Hahn~ a aa<:"CI ~ ~ Banach theorem, since A is equicontinuous, each xa can be extended by ya E- E 1 so that Ya(a E- V) is still equicontinuous. There exists hence y 0 E- E 1 adherent to y a( a E- €7) for the topology cr(E' ,E). Since y hence Ty
0
=
a
-
0 on E. ,y ~
o
IE
i
=
o.
In other words, y
o
E- E~, ~
0. But q(Ty) is continuous for cr(E 1 ,E) on every
equicontinuous subset of E',
thus 0 = q(Ty 0 ) is adherent to
q(Ty ). Hence q(T. x ) = q (Ty) is not larger thanE: for eacha. a ~ a a There are interesting examples of spaces F satisfying the assumptions of prop.V.1.6. DEFINITION v.1.7.
Let us call Q the class of all l.c.s. E such
that, for every sequence of continuous semi-norms q n (n E-JN), there exist a continuous semi-norm q and a sequence A. n ;:., 0 (n E-JN) such that qn(x) ~ "-n q(x),
\fn E-JN,
\fx E- E.
It is equivalent to say that, for every sequence U n (nE-JN) of there exist another neighborhood U
neighborhoods of zero in E,
of zero and a sequence t..n(nE-lN) such that
PROPOSITION V.1.8. The class Q contains a) the Banach spaces, b) ~ ~~-spaces and in particular the strong duals Fb of metrizable spaces F, c) the duals F' of metrizable spaces F and the duals F' _of c ca ~-spaces,
For a), it is obvious; b) is lemma 2.p.64 of [44]. 84
For c), to each qn' there corresponds a compact set Kn in F such that qn (x') ~
sup l<x,x'>l, Vx' ~ F•, Vn ~ m. x~Kn
If 11·0 fast enough (for instance, if U.(i~m) is a base of 1 1 neighborhoods of zero in F, take 11i such that 11i Ki C Ui), then 00
K = {0} U
U
i=1
11 i Ki
is compact in F and sup l<x,x'>l ~ _L sup l<x,x'>l 11n x~K
x~Kn
'
hence the result. If moreover F is complete, c;(K) is still compact. PROPOSITION V.1.9. The class Q is stable for a) subs paces, b) quotients, c) countable inductive limits and direct sums, d) finite products. a) If L is a vector subspace of E, a neighborhood of zero in L is the restrict1on to L of a neighborhood of zero in E, hence the result. b) If E/L is a quotient space and j
the canonical quotient
map, every neighborhood of zero in E/L is the image by j
of a
neighborhood of zero in E. Hence the conclusion.
c) Let E be the inductive limit of the spaces Ei (i ~lN) and Uk(k ~m) a sequence of neighborhoods of zero in E. There exist neighborhoods
U~~)
of zero_in Ei such that Uk )
U~i)
for each
There exist then A.~ 1 ) > 0 and a neighborhood Vi of zero
i ~lN. in each Ei such that
~ k=1
A.(i) k
u(i) ' k
85
Put 1/sup A(i)
lli
k
lc.!;;i
and
00
v = Co(
u
llivi).
i=1 Then uk
)
u(i) k
and
)
1
0TI k
v.l.
)
lli vi, Vi
:1!:
k
'
for £k small enough. Thus 00
uk) inf (1,£k) Co( u lliv 1 ), Ilk~ E i=1
,
00
where Co( U ll·V.) is a neighborhood of zero in E. i=1 l. l. d) is straightforward. The following simple argument will provide new examples of webbed spaces of linear maps. Let J be the map T -
T*. For sui-
table topologies on the various spaces involved, J may happen to be continuous and surjective of L(E,F) into L(F 1 ,E 1 reover L(E,F) is webbed, then L(F ,E 1
).
If mo-
is webbed too by prop.
1 )
IVe4o2o
In the sequel of this paragraph,
~.~'
denote any covering
family of bounded sets, which is stable under continuous linear maps, i.e. T~(E) C ~(F) for all T ~ L(E,F). PROPOSITION V.1.10. The map J is defined of L(E,F) ~ L(F~,E~) for all ~.
Indeed, if T ~ L(E,F), for every B ~~(E), TB belongs to the corresponding family ~(F). Thus sup l<x,T*y'>l = sup Il x~B
x~B
which yields the continuity of T* of
86
sup ll y~TB
F~
into
E~.
PROPOSITION V.1.11. The map J
is continuous of L~(E,F) into
LJ.3(F JJ' ,E~,) for all 9.3 .!!.!ll! 9.3 Indeed, to B ~ 9.3(F~ 1 ) and B• 1 •
~ 9.3'(E) corresponds
the neigh-
borhood of zero IT' ~ L(F.JJ, ,E.JJ,)
w=
sup
sup
x~B'
in LJ.3(F~ 1 ,E~ 1 -1
J
W
=
).
l<x,T 1 y'>l ~ 1}
y 1 ~B
Its inverse image by J,
IT~ L(E,F)
: sup xE-B'
sup
jj ~ 1}
y'~B
is clearly a barrel in L (E,F). Thus it absorbs every Banach s disk of Ls(E,F) and it is a neighborhood of zero in L~(E,F). PROPOSITION V.1.12.
I£ E is evaluable, the map
J
is surjective
££ L(E,F) ~ L(F~,E~) ~ t9.3 is coarser than t9.3
ill F•. 't"
let T 1 belong to L(F~,E~). For every y'
Indeed,
~ F•,
T'y'
is a linear form onE and, for each x E- E, y ' - <x,T'y'> is a linear form on F 1 • ce t9.3
~
(x,T y 1
1
t9.3 , >
't"
=
It is moreover continuous on F.JJ, hence, sin-
there exists a unique element Tx (Tx,y
1 )
~
F such that
for ally E- F'. We define this way a linear
map T of E into F such that T*
=
T•. The map Tis continuous.
Indeed, Tis continuous of Eo(E,E') into Fo(F,F•)• Thus it maps bounded subsets of E into bounded subsets of F (by the Mackey boundedness theorem). If V is a closed and absolutely convex neighborhood of zero in F, T- 1 v is thus absolutely convex, closed and bornivorous. Since E is evaluable, it is thus a neighborhood of zero. Combining the above results, we get the following properties. PROPOSITION V.1.13.
~
E be a Banach space or a countable in-
ductive limit of Banach spaces and F a strictly webbed l.c.s. strictly webbed if t9.3 1 is coarser then t9.3 't"
illF'• By prop.V.1.3, L~(E,F) is strictly webbed, By prop.v.1.11, the map J is continuous of t 13 (E,F). into LJ.3(F9J 1 ,E~, ). By prop. V.1. 12,since E is evaluable and t9.3 1 is coarser than t9.3 in F 1
,
't"
87
J
is moreover surjective. Conclude by prop.IV.4.2.
PROPOSITION V.1.14. Under the assumptions of prop.V.1.4, L~(F~ 1 ,E~ 1 )
is webbed or strictly webbed provided t~ 1 is coarser
ill F .. Same argnment.
.ih.!m t ~
I •
A rough application of the same ideas to prop.V.1.5 does not bring any new result. Indeed, if E is metrizable and E' evaluac
ble, it is equal to Eb and as we have seen in the proof of prop. III.).12,Eb is a countable inductive limit of Banach spaces. A more careful approach can however lead to some interesting results. PROPOSITION V.1.15.
E be metrizable and F admit a strict web
~
(resp. be sequentially complete and webbed). ~ Lr(F~a'E) ~ strictly webbed (resp. webbed), where r is the class of eguicontinuous subsets of F 1 T
1
•
The map J is sur,iective of L(E 1 ,F) into L(F 1 ,E). Indeed, let c ca belong to L(F 1 ,E). For each x 1 f El, ca y
I
-
(T
I
y t
t X I)
is an element of (F'ca ) 1 , hence of F. Define T by =
into F. Indeed, if V is a neighborhood of zero in F, V 0 is
equicontinuous, hence compact in F' , thus T 1 V 0 is compact in ca E and y
sup 1 fV 0
Il =
sup zfTIV 0
ll
1 is a continuous semi-norm of E c•
Moreover, Lr (F'ca ,E) sup y 1 fV 0
T' is continuous of Lb(E~,F) into if U and V are neighborhoods of zero in E and F, the map J
sup fU 0
X 1
: T
I<JTy 1 ,x'>l
sup sup fU 0 y 1 fV 0
Il
X 1
is a continuous semi-norm of Lb(E~ 1 F). The conclusion follows from prop.V.1.5.
88
2. TENSOR PRODUCTS We do not have any direct information about the permanence properties of webbed spaces with respect to tensor products. However tensor products can be looked at as subspaces of spaces of linear maps. When they are closed subspaces, it is thus possible to use §1 and prop.4.1. For the general theory of tensor products of l.c.s., we refer to [95]. PROPOSITION V.2.1. The tensor product E
~
E
F of two l•c.s. is
a subspace of Lr(E~a'F), equipped with the induced topology, where r is the family of eguicontinuous subsets of E•. Moreover Lr(E'ca ,F) is a closed subspace of Lr(E ~1 ,F). It is well known from algebra that the map x ® y where
Tx ® y'
Tx ® y x' = <x,xl)y extends to a linear injection of E ® F into L(E 1 ,F), provided E 1 is equipped with a topology t such that (E 1
~
By definition, the topology
E
of E
~
) 1
=E.
F is the topology of
uniform convergence on equicontinuous sets of Et and F 1
,
so it
is the topology induced by Lr(E~,F). LetT belong to L(E 1 ,F). We have ~
T E- L ( E 1
ca
,
F)
<=>
T * E- L ( F
1 ca , E ) •
Indeed, if T E- L(E~a'F), then T* E- L(F~,(E~a)~). But (E~a) 1 = E and, for 9J = 9J , the topology of (E 1 )~ is finer than the inica ca ~ tial topology on E. Conversely, if T* E- L(F~a'E), (T*)* coincides with T hence, by the same argument, T E- L(E~a'F). Let now U be a closed and absolutely convex neighborhood of zero in E. We have T*-1
u
{y I E- Fl {y I E- Fl
I
I , X
I> I
Il
~
~
1'
lfx 1 E-
1' lfx 1 E-
u•} u•}
(Tu•)•.
89
SoT* E- L(F' ,E) if and only if TU• is compact in F for each u. ca Since u• is compact in E' , this is the case if and only if the ca restrictions of T to the equicontinuous subsets of E' are continuous for the given topologies on E 1 and F. Assume now that T belongs to the closure of L(E~a'F) in L (E',F). By definition of r, for each neighborhood U of zero
r
in E, T is the uniform limit on u• of elements of L(E~a'F), which are continuous in u•. SoT is also continuous in u•, hence T E- L(E'ca ,F). The completion E ®
£
F is important for applications. It is a
closed subspace of Lr(E~,F) if Lr(E~,F) itself is complete. PROPOSITION V.2.2. l l E and F are complete Lr(E~,F) is complete. Let Ta(a E- U) be a generalized Cauchy sequence in Lr(E~,F). For each x 1 E- E•, T x' is a generalized Cauchy sequence of F. Its a
limit T x' defines a linear map T of E' into F. 0 ~ ~ Let us prove that T 0 is continuous of E' in F. First, it is continuous of E~ into F equipped with the topology o(F,F
1 ).
Indeed, the map T - T* is an isomorphism between Lr(E~,F) and Lr(F~,E)
(just write down the basic neighborhoods of zero).
Thus, for each y' E- F•, T*y' is a generalized Cauchy sequence a and converges to some z E- E. Hence lim a
lim • a
and x ' - is continuous in 1
E~.
The adjoint T: of T 0 is also continuous ofF~ into Eo(E,E•)• If V is a closed and absolutely convex neighborhood of zero in F, T~v· is compact and absolutely convex in Eo(E,E•)• Thus (T*0 V•)• is a neighborhood of zero in E' and, since T (T*V 0 )° CV, ~
T 0 is continuous of
E~
0
0
into F.
PROPOSITION V.2.,. l l E is an
~-space
and F a complete and A
webbed (resp. strictly wabbed) space, ~ E ® F is webbed £ (resp. strictly webbed).
90
1£
E is the strict inductive limit of a sequence of
~-spaces
and ifF is complete, belongs to the class Q (cf.def.V.1.7) ~
..
is webbed (reap. strictly webbed), then E ®
£
F is webbed (reap.
strictly webbed). In both cases, every compact subset of E is contained in an absolutely convex compact subset of E. Thus, by prop.V.1.5 and 6, Lb(E'ca ,F) is webbed (reap. strictly webbed). It is obvious 1 1 that Lb ( E ca , F) = Lr ( E ca , F ) • s o Lr ( E ' ca , F ) is a webbed (reap. strictly webbed) space. It is moreover complete, by prop.V.2.1 and 2. Thus E ®
£
F is
a closed subspace of Lr(E'ca ,F) and it is webbed (reap. strictly webbed). PROPOSITION v.2.4.
. ®
. ®
~
E be an V-space or a countable inductive
limit of v-spaces and F be a complete and webbed l•c.s. E' ca
£
F and Eb' ---
£
~
F are webbed.
They are strictly webbed if - either E is a Banach space or a countable inductive limit of Banach spaces and F is strictly webbed, - ~' with no extra assumption on E, i£ F has a closed and absolutely convex web. Since E is ultrabornological, E£c is complete (prop.III.2.9) and, a fortiori, E' • Moreover, (E' ) = E, since every equica ca ca continuous subset of E 1 is relatively compact in E' and converca sely every compact disk in E' is equicontinuous. Since F is ca .. complete too, L((E' )',F)= L (E,F) is complete and E' ® F ca ca ca ca £ is a closed subspace of it. The conclusion follows then by prop.V.1.3 and 4. On the other hand, it is easy to verify that Eb ®£ F is a subspace of Lb(E,F), equipped with the induced topology. Since F is complete and E bornological, Lb(E,F) is moreover complete. Hence the result.
91
3. SPACES OF VECTOR VALUED FUNCTIONS One of the main applications of the theory of topological tensor products is the study of spaces of vector valued functions. For instance,
the space &(E) of infinitely differentiable functions
on mn with values in a l.c.s. E is isomorphic to a subspace of ~
&®
£
E (where & is the space of infinitely differentiable sea-
lar functions). When E is complete, &(E) is isomorphic to
&~
£
E hence, for the search of webs in &(E), we can use §2.
However the completeness assumption on E can be relaxed in many cases by a straightforward approach. DEFINITION v.3.1. The space E is said to be fast complete if every bounded subset of E is contained in a bounded Banach disk.
1£1
PROPOSITION V.3.2.
E be a l.c.s.
contained in a bounded Banach disk,
1£
B is bounded in E
~
there exists a smallest al-
ffebraically closed bounded Banach disk containing B. It is denoted c;(B) and given by 00
{ L:
C
m=1 Recall
m
X
x m ~ B, Vm
L:
m
m=1
-
f;
JN} •
that B is the algebraic closure of B •
Let B 0 be a bounded Banach disk containing B. If p is the gauge of B0
it is well known that
,
{x : p(x)
< 1} C B 0 C {x : p(x)
~ 1}
and thus B
{x : p(x)
0
~ 1}
is closed for tB • 0
If
X
m
f;
co
co
B for each m
f;
lN and
L:
L:
m=1
m=1
c
m
x
m
is con-
00
vergent in EB , hence in E. Moreover, E o m=1 co
B
1
= {
L:
B0
co
C
X
m m m=1 is contained in B0 •
92
em xm f
E
m=1
x
m
f
B,
Vm f lN}
•
Thus
Let us prove now that EB 1 isa Banach space. It is true if
( ""~
""
i=1 Each yi has the form
""~
( ""~
c.J.,m x.J.,m
m=1
m=1
~ A.i c.J.,m x.J.,m is absolutely convergent in the i,m Banach space EB • If xis its limit, it is clear that x f B 1
The series
N X
-
•
0
Moreover,
""
00
E i=N+1
E Ai yi i=1
A.i
E m=1
00
E
i=N+1
00
E A.i yi converges to X in EB I . i=1 00 I f X f B I ' we have X= E 2-mxfB 1 by the above argument. m=1 Thus B 1 is an algebraically closed Banach disk, hence the conclusion.
Thus
DEFINITION V.3.3. Let e be a set and E a l.c.s. We denote by L00 (e,E) the set of all maps ~ of e into E such that ~(e) is bounded in E, equipped with the topology of uniform convergence on e, defined by the semi-norms = sup p[~(x)] xfe where p is an arbitrary continuous semi-norm on E. PROPOSITION V.3.4. l f E is fast complete and webbed (resp. strictly webbed), ~ L00 (e,E) is webbed (resp. strictly webbed). Let fJl = {e
v
: v f
I}
be a completing (resp. strict) web of E.
Take it like in prop.IV.1.6. We can moreover assume that each e is algebraically closed replace each e by e From v v v
.
00
00
u A.l. i=1
A
=>
u A.l. i=1
A
and 00
00
A
)
~
i=1
A.i Ai
=>
A
)
~
i=1
A.i A.l.
93
it follows easily that
{e v :
v f
I}
is a new web which is still
completing or strict. For each~ f
L00 (e,E), C~ ~(e) is a bounded Banach disk of E. or 3 to the identical map of the corres-
Applying prop.IV.6.1
ponding Banach space into E, we get, -
there exists v f
- i f Co ~(e) C ev' and, when -
when~
is strict,
I such that C~ ~(e) C ev'
>
there exists v 1
v such that Co ~(e) C ev''
is only completing,
~
there exists v f
I such that e
n Ec; ~ (e) is non meagre in
v
Ec~ ~(e) and c; ~(e) C Co ev -
if the preceding statement is true for v, it is true for some
>
\1 I
V o
Assume that ~ is strict. Then ~·
= {&
v
I} , with
v f
Co ~(e) C e } v
is an absolutely convex web of L~(e,E). Given a strand &
\lk
(k flN),
let A.k(k ElN) be strictly associated
to the corresponding strand of to & X
f
(k ElN). Indeed, given Ilk f
vk e,
00
I t is also strictly associated
~.
[o,>..k] and
~k
f
&
vk '
for each
Ilk ~k(x) is convergent in E and
~
k=1 ~
!
~
k=k +1
Ilk ~k(x)
x
f
e}
C
0
00
By prop.IV.1.7,
I:
k=1 limit of
N I:
k=1
Ilk
~k
Ilk
~k
is thus an element of L00 (e,E) and the
for the topology of L00 (e,E). Thus ~·
completing web. We have moreover
We must improve this and show that
94
is a
co
co
( I: k=k 0 +1
c
Ilk IPk)(e)
e
,
vk
lfk 0 flN
0
We have co
co
11k c"'o q~k(e)
I:
c
Given C
>
k=N
e
VN-1
,
>
liN
1 •
neighborhood U of zero in E, there exist
~arbitrary
0 and N
c
Ilk e vk
I:
k=N
k +1 such that e C CU. Hence o VN
~
N E 11k Co IPk(e) + cu k=k +1 0
co
E 11k c~ IPk(e) k=k +1 0
c
c
c•u
for C 1 large enough and co
,...
Ilk Co IPk(e)
I:
k=k +1 0
is bounded in E for each k 0 f
~.
It is thus contained in a
Banach disk B. co
co
Given x f Co ( I: Ilk IPk)e, we can write it x = I: Am xm' k=k 0 +1 m=1 co
with x
co
I:
m
k=k +1
Ilk xk
0
,m
(xk
,m
f
IPk(e),
I:
m=1
IAmi
~ 1 ). The
series co
co
I:
I:
m=1 k=k +1
A
m
11
X
k
k,m
0
is absolutely convergent in EB. Thus co X
=
co
co
I: Ilk Co IPk(e) • Ilk ( I: Am xk ,m ) f k=k +1 k=k 0 +1 m=1 0 I:
Therefrom, 00
00
Co ( I: Ilk IPk)(e) k=k 0 +1
c
I:
k=k +1
Ilk Co IPk(e)
0
00
c
I:
k=k 0 +1
Ilk e vk
c
e
,
vk
Ilk
0
flN
.
0
95
When
~
is only a completing web, the web
~·
is the collection
of {cp E- L00 (e,E)
e v n E"" Co cp(e) is non meagre in Ec""o m(e) T an d cp ( e ) C C o ( e ) } • v
The proof is similar to the first part of the above one but
~·
is no longer strict. DEFINITION V.3.5. Let e (m E-lN) be an increasing sequence of subm sets of e. We denote by Loo (e m(mE-lN),E) the set of all maps cp of e into E such that cp(e ) is bounded in E for all m E-lli, equipped m with the topology of uniform convergence on the subsets e (m E-lN~ m COROLLARY V.3.6. !£ E is fast complete and webbed (resp. strictly webbed), .l!:!2 L00 (em(m E-JN)'E) is webbed (resp. strictly web~).
Indeed, L00 (em(mE-lli)'E) is the projective limit of the spaces L (e ,E) (mE-lN). Apply then prop.Vo3o4 and IV.4.7. "" m REMARK V.3.7. In prop.V.3.4 and cor.V.3.6, the assumption that E is fast complete is only used to assert that cp(e) is contained in a Banach disk. So both would hold true without the fast completeness of E, for the subspaces L:,(e,E) and L:,(em(m E-JN)'E) of all maps cp of e into E such that cp(e) (or cp(e )) is contaim
ned in a bounded Banach disk of E (for each m). There are many interesting examples of subspaces of L00 (e,E) or L00 (em(m E-IN)'E). DEFINITION V.3.e. We denote by l(E) the set of all sequences x (m E-lN) of E, equipped with the topology of pointwise converm
gence, defined by the semi-norms n(x ( E-lN)) m m
=
sup p(x.) , i~N
where p is a continuous
96
~
~emi-norm
of E and N E- IN.
The space l~(E) is the space of all bounded sequences, equipped with the topology defined by the semi-norms n(x (
LJN)) = sup p(x ) , mE-lN m
mm~
where p is a continuous semi-norm of E. The space c 0 (E) is the subspace of l~(E) of all sequences convergent to zero. PROPOSITION V.3.9. ~ E be webbed (resp. strictly webbed). ~ l(E)
is webbed (resp. strictly webbed).
l!
E is moreover
fast complete, l~(E) ~ c 0 (E) are webbed (resp. strictly webbed). The space l(E) is l 00 (em(mE-lN)'E), where em= For each 4l E- l,)em(m
e-m) ,E),
{1, ••• ,m).
4l(em) is finite, hence contained
in a bounded Banach disk. The conclusion follows from cor. V.3.6 and remark V.3.7. The space 1 00 (E) is 1 (lN,E) and c 0 (E) is a closed subspace ~
of 1 00 (E). Regarding spaces of continuous or differentiable functions, we will restrict our investigations to two striking examples. Many others can be handled by the same arguments. DEFINITION V.3.10. Let E be a compact space and E a l.c.s. We denote by C(K,E) the space of all continuous maps of E into E, equipped with the topology of uniform convergence on K. Let Q be an open subset of mn. We denote by 8(Q,E) the space of all infinitely differentiable maps of Q into E, equipped with the topology defined by the semi-norms sup sup p(D~ 4l) xE-K jo:j.e;m where E is compact in Q, p a continuous semi-norm of E and Do: X a derivative of~ of order
jo:j.
97
!p
PROPOSITION V.}.11. lL E is fast complete and strictly webbed (resp. webbed) ~ C(K,E) ~ &(a,E) are strictly webbed (resp. webbed). The space C(K,E) is a closed subspace of Lco (K,E). If Km(m E-lN) is a increasing sequence of compact subsets of Q such that K C K0 1 for each m E-m, &(a,E) is isomorphic to a closed m m+ subspace of the product of countably many copies of L00 (Km(m E-li:{)'E), through the embedding which, to each II' E- &(a,E), associates the sequence of all its derivatives (arranged in a given order). Various spaces of linear maps can also be recovered by similar arguments. PROPOSITION V.}.12. 1Al E be a normed space or a countable inductive limit of normed spaces Ei (i E-lN) ~ F be fast complete and webbed (reap. strictly webbed). ~ L~(E,F) is webbed (reap. strictly webbed).
lL E or the E.'s are Banach spaces, F does not need to be l. fast complete. It is an improvment of prop.V.1.}. Let B (reap. Bi) be the closed unit ball of E (reap. Ei). When E is normed, the map J : T- TIB is an isomorphism of Lb(E,F) onto a closed subspace of L~(B,F). If E is the inductive limit of the normed spaces E.(iE-IN), J aT -TI 00 is l. U Bi i=1 an isomorphism of L~(E,F) onto a closed subspace of L00 (~,F), where ~ = {Bi : i E-lN}. Indeed, J is clearly injective. Moreover, J L(E,F) is the set of all II' E- L00 (~,F) such that ~p(
N I:
i=1
ci xi)
=
N I:
i-1
ci ~p(xi)
00
for all N E- 1N' xi E-
u
j=1
Bj and ci such that
is trivially closed in L00 (~,F).
98
N I:
i=1
Ic 1 I
~ 1.
It
Thus, by prop.V.3.4 and cor.V.3.6, Lb(E,F) (or L~(E,F)) is webbed or strictly webbed. If
~·
is the web defined in it
according to prop.V.3.4 or cor.v.3.6, for each strand 8 and each associated sequence A.k(k E-lN), the series
vk
(k E-lN)
N ~
Ilk Tk k=1 (Ilk E- [O,A.k], Tk E- &vk) (n E-lN) are equicontinuous. Arguing- as
in the proof of prop.V.1.3, we obtain
that~·
is a completing
or strict web of L~(E,F). If E is a Banach space or an inductive limit of Banach spaces, use remark V.3.7. The improvment of prop.V.1.3 could be derived by a simple modification of its proof : the main step is that, since B is a bounded Banach disk, prop.IV.6.1 can be used to claim that TB is absorbed by elements of the web of F. If E is for instance only normed but F fast complete, C~o(TB) is again a Banach disk and the proof can be continued with Co(TB) instead of TB. The same argument also provides another proof of prop.V.1.5: using the notation of prop.V.1.5 and its proof, Lb(E~,F) is isomorphic to a closed subspace of L00 (u;(m E-lN)'F). 4• LOCALIZATION PROPERTIES FOR SUBSETS OF L(E,F) Let E be an
~-space
~-spaces Fi(iE-lN).
and F the inductive limit of a sequence of It is well known (cf.cor.IV.6.5) that, if T
is a continuous linear map of E into F, then TE C Fi for some i
E- lN. If we replace T by a subset ~ of L(E,F), under which
conditions is it true that, for a fixed i
T E-
E- lN, TE C Fi for all
~?
The question
~as
been asked by Hirschfeld and solved by
Kothe in [65]. Further investigations are due to Floret [39]. We have seen in chapter IV, §6 that the above property, for a single map, is widely generalized by the localization property (prop.IV.6, 1-2-3). So it is natural to ask a similar question for webbed spaces. The description of the webs of spaces of linear maps provides an easy answer.
99
PROPOSITION V.4.1.
=
~ ~
E be a Banach space and F have a strict
~
{ev : v ~I}.
l£
B is the closed unit ball of E ~ $
a bounded Banach disk of L (E,F), s - there exist v
- iL
~
I
~
C
> 0 such that
$B C Ce , there exist v'
) v
v
~
~B
>
C1
C Cev 0 such that
$B C C 1 e ,. v
- in particular, there exist a strand e l l Ck
>
0 ( k ~ lN) such that
Let us first modify the
vk
(k ~
m)
and a sequen-
for each k ~ IN. vk according to prop.IV.1.6 in
B C Ck e
web~
order to get rid of the constants. By prop.V.1.3, ~
= {8 v
1
8
v
: =
v ~ I} , where {T ~ L(E,F) : TB C e } v
is a strict web of Ls (E,F). The conclusion follows from prop.IV.6.1, applied to the identical map of E PROPOSITION V.4.2.
~
into L (E,F). s
~
E be normed and F fast complete and
equipped with a strict web
~
=
{e
v
: v ~ I}.
l£
B is the unit
ball of E ~$a bounded Banach disk of L (E,F), there exist s a strand ev (k E- lN) and Ck > 0 (k E- IN) such that k
for each k ~IN. vk Same argument, using this time the web of L (E,F) provided s by prop.Vo3o12.
$B C Ck e
COROLLARY Vo4•3• In both prop.V.4.1
~
2, assume that F
~
the inductive limit of a sequence of strictly webbed spaces Fi(i E-IN). Then there exists k ~lN such that $E C Fk and that $
is equicontinuous in L(E,Fk).
i
Equip F with the web of prop.IV.4.5 1 e{i} = Fi for all ~lN. It is then immediate that $B (hence $E) C Fk for some k.
The equicontinuity follows from the fact that $
is then bounded
in Lb(E,F), hence equicontinuous (since E is normed).
100
PROPOSITION V.4.4.
~
E be a Baire space and F be fast com-
plete and admit a strict web~= {e : v ~I}. Denote by U A v base of neighborhoods of zero in E.
l!
~
(k ~ m) vk for each k ~lN.
is bounded in L (E,F), there exist a strand e s
and a sequence Uk(k ~ lN) .iJl U such that ~Uk C ev k
We assume again
that~
is modified according to prop.IV.1.6.
This time, there is no web in L(E,F) that could be used in the proof. We proceed as for the closed graph theorem. For each x ~ E, ~x is bounded in F, hence Co(~x) is a bounded Banach disk of F. Applying prop.IV.6.1 map of F ~·
~
Co
(
~x
to the identical
) into F, it is easy to see that the collection
of g
(v ~ I) C0(:1.3x) ( e } = {x ~ E v v is an absolutely convex web in E. Thus there exists a strand g is non meagre for each k ~ m. (k ~ IN) of fl?. 1 such that g vk vk Let Ak(k ~m) be strictly associated to the strand e
vk
(k ~IN). There exist Uk f
U(k E-IN) such that
A fortiori,
By the proof of the closed graph theorem (prop.IV.5.1) adapted to the case when E is a Baire space (prop.IV.5.3), it follows that Uk C T
-1
e
vk-1
, Ilk
~
m,
'IT E-
~
henne that , Ilk E-IN. Hence the result.
1 01
PROPOSITION Vo4o5o
~
E be metrizable and F be fast complete
and admit a strict web~~ {ev : v f
I}.
1L
~is
eguiconti-
nuous in L(E,F), - there exists v f
- i!
~E
C >e
v
<,
For each v,
I such that
~E
C >ev< ,
there exists v' > v such that {me
v
1
:
~E
C )e
v
,< •
m f lN, v' > v} is a strict web of
)e (. Thus, it is enough to prove that there exists v ~~ v
such that
~E
C )e
v
<•
If it is not true, there exists for each
i ~lN, xi f E and Ti f ~such that Tixi f )e{iJ<• Since E is metrizable, there exist constants c. > 0 (i flN) such that l.
ci xi - 0 in Eo Since ~ is equicontinuous, ci Ti xi - 0 in F. Thus {ci Ti xi : i fm} is contained in a bounded Banach disk B of F. Applying prop.IV.6.1, to the identical map of FB into F, B is absorbed by some e{i}' hence a contradiction.
102
VI Homomorphisms and weak homomorphisms 1. HOMOMORPHISMS AND DUALITY DEFINITION VI.1.1. Recall from def.I.}.1 that a linear map T of E into F is open if Tw is open in F for every open subset w of E. An open map is thus onto. We say that T is relatively open if it is open of E into R(T), thus if Tw is open in R(T) for every open subset w of E. It is weakly open (resp. relatively weakly open) if it is open (relatively open) when E and Fare equipped with the weak topologies a(E,EI) and a(F,FI). We have also seen (prop.I.2.5) that, if the graph of T is c 1 o sed in E x F , ~ ( T*) ( i o eo {y I E- F : T*y I E- E I } ) is a ( F I , F)dense in Fl. PROPOSITION VI.1.2. LetT be a linear map of E into F.~ ker T* = R(T) 0 ~ R(T*) 0 ) ker T. If the graph of T is closed ~ E x F, ~ ker T = R(T*) 0 , hence (ker T) 0 is the a(EI,E)closure of R(T*). Indeed, y 1 E- R(T) 0 (~ (Tx,yl) = O, Vx E- E <~
T*yl
~
<x,T*yl) =
~
=
0
0
Moreover, X
E- R(T*)
0
)
o, o,
Vy I E- ~(T*) Vy I E- ~(T*)
The last condition is true i f X E- ker T and equivalent to X E- ker T i f ~(T*) is a(F•,F)-dense in F I 0
PROPOSITION VI.1.}. ~ T be a linear map of E into F. It is relatively open if and only if, for every continuous semi-norm p ~ E, the semi-norm
1 0}
q(Tx)
inf
p (X+ X I )
x 1 ~ker
T
is continuous on R(T). Indeed, T is relatively open if and only if every absolutely convex neighborhood U of zero in E is mapped onto a neighborhood of zero in R(T).
If p is the gauge semi-norm of U,
the
gauge semi-norm of TU is q, hence the result. PROPOSITION VI.1.4.
T be a linear map of E into F;
~
a) T is relatively weakly open if and only if R(T*) = (ker T)
0
,
b) i f G(T) is closed in E x F, T is relatively weakly open if and only if R(T*) is o(E•,E)-closed. a) By prop.VI.1.2, we have ker T C R(T*) R(T*) C (ker T)
0 •
Take now y ~ (ker T)
<x,y> if Tx
<x',y'>
=
x'
0 •
thus
0 ,
Define y'
on R(T) by
•
The definition makes sense since Tx Now y'
=
Tz => x - z
ker T
~
) <x,y> = •
is a linear form on R(T). If T is relatively weakly open
or relatively open, y' [x' ~ R(T):
is continuous : indeed,
j<x',y'>l < 1} = T({x~E: j<x,y>j < 1})
is open in R(T). By the Hahn-Banach theorem, y' is the restriction to R(T) of a continuous linear form y" on F. From
=
(x,y>,
wx
~
E
it follows that y" ~ ~(T*) and y Conversely, assume that R(T*) that Tis relatively weakly open, ~
E
x 1 ~ker
T
for all x,, ••• q(Tx) =
,x~
inf
T*y" ~ R(T*). (ker T)
0
•
We must prove
or, by prop.VI.1.3, that,
1 ,
sup l<x+x 1 ,xi>I i~n
is continuous on R(T) endowed with the topology induced by a(F,F•). Choose a maximum number of x!1 such that xil
1 04
are ker T
linearly independent. Assume that these are xi(i~p). For suip
table cij' Yj = X! -
l;
J
c
i=1
0
0
~J
x! = 0 on ker T for j
> p. It is
~
clear that x!(i~n) belongs to the linear hull of x!(i~p) and ~
~
yj(j=p+1, ••• ,n), so that, up toaconstant, q(Tx) is less than the se!"li-norm q 1 (Tx)=
inf
x'~ker T
sup (sup l<x+x•,x!>l, i~p
l<x+x',y!>l)•
~
J
There exist xi(i~p) in ker T such that (xi,xj>=oij(i,j~p). Then, for each x
E,
~
p
q 1 (Tx)~sup(sup i~p ~sup p ~~n
<
0
p
l<x- l: <x,x!>xo,x!>l, sup l<x- l: (x,x!>xo,y!>l) j=1 J J ~ p(i~n j=1 J J ~
l<x,y!>l ~
Since y!~ ~ (ker T)
0
= R(T*), there exist z!~ (p
F such
that T*z!~ = y!, hence ~ q(Tx) ~
sup
p(i~n
Il, Vx ~ E , ~
where the right hand side is a continuous semi-norm of R(T) for the a(F,F
1 )
topology. Hence the result.
b) If G(T) is closed, by prop.VI.1.2, R(T*) (ker T)
0
0
= ker T, hence
is the closure of R(T*) in E~(E•,E)" Hence the result.
PROPOSITION VI.1.5. A linear map T
~
E
~
F is almost open
if and only if every eguicontinuous subset of R(T*) is the image by T* of an eguicontinuous subset of F 1
•
Let T be almost open. If A is equicontinuous in R(T*), there exists a neighborhood U of zero in E such that A C R(T*) n U 0
•
Since T is almost open, there exists a neighborhood V of zero in F such that TU ) R(T)
n v.
We may assume that V is open and absolutely convex. Therefrom, R(T)nV C TU
(*)
1 05
Indeed, since V is open, iTT) n V C R(T)nV. Moreover, if X
RtTT RtTT
f
X f
R(T)nv
c
n
v,
n v
RtTT
we have
c
RTTJ
f
AX
n v for all A f
[0,1[, hence
R(T)nv. Conversely, it is obvious that n
v.
RTTT
Taking the polars,
n
v c TU
yields
since V0 is o(F•,F)-compact, hence R(T)
+ V0
0
is o(Ft,F)-closed.
On the other hand, {y f F'
ll ~ 1, "'x f
:
{y f ~(T*) : T*y f
U0
U}
= T*- 1 U0
}
o
Thus
and, since R(T) 0 U0 n R(T*)
= ker T*, = T* T*- 1 U°
C T* V 0
•
Therefrom, A
= T*(V 0 n T*- 1 A)
•
Conversely, assume that, for every neighborhood U of zero in E, there exists a neighborhood V of zero in F such that R(T*) n U° C T* V 0
•
We have (TU)
0
= T*- 1 uo
C vo + ker T*
V0
+ R(T)
0
and, taking the polars, TU ) R(T) n V if U is absolutely convex.
PROPOSITION VI.1.6.
A
linear ma.p T ,!ll. E i.!:!.i2 F is relatively
open if an only if it is almost open and relatively weakly open. If T is relatively open, it is clear that it is almost open. Moreover, the proof of prop.VI.1.4, actually shows that T rel. open ~ R(T*) 106
= (ker T)
0
<=9 T rel. weakly open.
Conversely, assume that T is almost open and relatively weakly open. By prop.VI.1.5, for every neighborhood U of zero in E, there exists a neighborhood V of zero in F such that T*V 0
U0
)
n
R(T*)
o
We may assume that U and V are closed and absolutely convex. Take the polars of both members :
{x
~
E
l<x,T*y'>l ~ 1, ~yt ~ vo}
{x
~
E
Tx ~ yoo} = T-1V
and, by prop.VI.1.4.a),
Hence T- 1 v
<
and T U )
t
2U + ker T V n R(T) •
DEFINITION VI.1.7. A linear map T of E into F is an isomorphism (reap. a weak isomorphism) if it is continuous and has a continuous inverse, or, in other words, if it is continuous, open (which implies that it is surjective) and injective (reap. the same, with the weak topologies cr(E,E 1 PROPOSITION VI.1.8. a) R(T*)
=
~
)
and cr(F,F•)).
T be a linear map of E
~
F;
E' if and only if T is injective and relatively
weakly open. b)
1L
G(T) is closed in E x F, R(T)
= F if and only if T* iA
injective and relatively weakly open of ~(T*) ~ E•. c) T is a weak isomorphism of E weak isomorphism of F•
~
~
F if and only if T* is a
E1 •
a) By prop.VI.1.4, if T is relatively weakly open and injective, then R(T*)
= (ker T)
Conversely, if R(T*) ker T
<
R(T*) 0
=
0
and ker T
= {o},
thus R(T*)
= E•.
Et, by prop. VI.1.2, we have
= {o}, hence Tis injective. Moreover 107
R(T*)
(ker T)
0 ,
thus, by prop.VI.1.4, T is relatively weakly
open. b) If G(T) is closed in Ex F, ~(T*) is o(F•,F)-dense in F• (prop.I.2.5). The dual of ~(T*) equipped with the topology o(F•,F) is thus (isomorphic to) F. Similarly,
(E!)' = E
and
= T. Thus b) is a particular case of a).
(T*)*
c) Let T be a weak isomorphism of E into F. Then ~(T*) = Fl and T* is continuous of F 1 into E 1 • Moreover, G(T) is closed s s and R(T)
=
F thus, by b), T* is injective and relatively open.
Finally, Tis injective and weakly open, hence R(T*)
= E1 •
Thus T* is a weak isomorphism. For the converse, identify
E
to
(E')•, s
F to (F')' and T to s
(T*)*. PROPOSITION VI.1.9.
.i..!l!.2
~
T be a continuous linear map of E
F•
a) It is an isomorphism of E into R(T) if and only if every eguicontinuous subset of E' is the image by T* of an eguicontinuous subset of F•. b) 1! F is a Mackey space, ~ T is open if and only if it is weakly open. a) Indeed, by prop.VI.1.6, Tis an isomorphism of E into R(T) if an only if it is injective (~ ker T ..
{o}
<=>
(ker T)
it is relatively weakly open (~ ) R(T*) prop.VI.1.4),
=
0
=
E' ),
(ker T)
0 ,
by
- it is almost open. The two first conditions are equivalent to R(T*) = E• and, by prop.VI.1.5, the last one combined with R(T*) = E• is equivalent to the condition given in a). b) If T is weakly op~n, it is onto hence, by prop.VI.1.8,b), T* is a weak isomorphism of F• into R(T*). Moreover, by
108
prop.VI.1.4,a), R(T*) is a(E•,E)-closed. If U is a neighborhood of zero in E, U0 n R(T*) is absolutely convex and compact for a(E 1 ,E). Thus T*- 1 (U 0 n R(T*)) is also absolutely convex and compact for a(F•,F). Since F is a Mackey space, it is thus equicontinuous. By prop.VI.1.5, Tis thus almost open and, by prop.VI.1.6, it is open. The converse follows immediatly from prop.VI.1.6. When we particularize E and F to the class of
a
~-spaces,
number of conditions turn out to be equivalent. PROPOSITION VI.1.10. linear map of E
ilii£
E,F
~
~~-spaces
and T a continuous
F. The following conditions are eguiva-
~:
(a) T is relatively open, (b) T is relatively weakly open, (c) T* is relatively weakly open, (d) R(T) k..s:losed in F, (e) R(T*) is closed in E' ~ a(E•,E), (f) R(T*) is fast sequentially closed in E' ~ a(E•,E).
>
~
a
b
It is a consequence of prop.VI.1.9,b), since R(T) is metrizable, hence a Mackey space. ~>
b
e
A consequence of prop.VI.1.4,b). ~
e
>
f
The implication e
=>
f is obvious. Let us prove that f
=>
e.
By the Krein-Smulian theorem (prop.I.2.7), R(T*) is closed in E' for a(E•,E) if, for every neighborhood U of zero in E, R(T*)
n U
0
is closed for a(E•,E). By assumption, R(T*)
n
uo is
closed in the Banach space Euo• Hence it is a Banach disk. Let Vm(m E-IN) be a base of neighborhoods of zero in F. Then 00
F
I
=
u V 0 and m m=1
R(T*)
n
U° C
00
U T* m=1
v; • 1 09
0 Since T* is weakly continuous, each T*V m is compact hence closed in E 1 for a(E•,E). Thus T* n Euo is closed in the
v;
Banach space EuonR(T*)• By prop.I.1.4, there exists m that T* v; n Euo is a neighborhood of zero in Euo from, there exists A > 0 such that
~
m such
n R(T*)•
There-
0 U0 n R(T*) C AT* Vm
and U0 hence U0
n
R(T*)
n
0 = U0 n AT* Vm
R(T*) is closed in Et for a(Et,E).
c <=9 d Another consequence of prop.VI.1.4,b) by identifying E to (E~) 1 , F to (F~) 1
and T to (T*)* and observing that R(T) is
closed in F if and only if it is closed in Fa(F,Ft)• a
<=>
d
The implication d => a follows from the open mapping theorem (prop.I.3.2). For the converse, observe that T induces an isomorphism between E/ker T and R(T). Since E/ker T is complete, R(T) is also complete, hence closed in F. Condition (f) is, as far as we know, due to Edwards in a slightly different form [35, thm 8.6.13]. The material above is widely inspired by [35, §a.6]. 2. SURJECTIVITY THEOREMS
The results of §1 provide various conditions for the surjectivity of T and T*, by observing that R(T) = F is equivalent to R(T) is closed and dense and that R(T) is dense if and only if T* is injective. First, prop.VI.1.8 characterizes the surjectivity ofT and of T*. For ~-spaces, we can particularize prop.VI.1.10. PROPOSITION VI.2.1. ~ E ~ F ~~-spaces and T a continuous linear map of E ~ F. The following conditions are equivalent: 110
(a) T is sur,jective, (b) T is open,
(c) T is weakly open, (d) T* is a weak isomorphism of F• ~ R(T*), (e) T* is injective and R(T*) is fast sequentially closed in E1
.£..!2£ cr(E•,E).
(f) R(T) is closed nnd T* is injective. Straightforward consequence of prop.VI.1.10. PROPOSITION VI.2.2.
~
E
~
F
~~-spaces
and T a continu-
ous linear map of E into F. The following conditions are equivalent (a) T* is surjective, (b) T* is weakly open, (c) T is a weak isomorphism of E ~ R(T), (d) T is an isomorphism of E ~ R(T), (e) R(T) is closed in F ~ T is injective, (f) R(T*) is closed (or fast sequentially closed) ~ E' for cr(E 1 ,E) ~Tis injective. By prop.VI.1.2, R(T*) is dense in E' if and only if R(T*) s
0
=
ker T = !O}. To be equal toE', it must moreover be closed in E'. The result is then a simple consequence of prop.VI.1.10. s
The next result is inspired by Treves [104,
thm 37,2] quo-
ted in prop.VI.2o4o PROPOSITION VI.2.3. linear map of E
~
~
E,F
~
~-spaces
and T a continuous
F. Assume that, for every neighborhood U ~ F such that
of zero in E, there exists a vector subspace N R(T) + N ~ R(T)
=
F
~
uo n
R(T*) C T*N° (resp.T*
is closed in F (reap. R(T)
=
-1
U° C N°).
F).
It is clear that T*- 1 U° C N° ( ) U0 n R(T*) C T*N° and ker T* C N° Since ker T* 1 11
ker T* C N° (~ N C
RTTJ • F, it implies that R(T) is dense in F.
When moreover R(T) + N
So it is clearly enough to prove the first assertion. We may assume that N is closed. The space F/N is still an ~-space.
If n is the quotient map, n o T is continuous of E
= F/N is closed in F/N. By prop.VI.1.10,
into F/N and R(noT)
R[(noT)*] is thus a(E•,E)-closed in E•. By the canonical identification of (F/N)t and No, (noT)* corresponds to T*INo• Thus T*N° is a(E 1 ,E)-closed in E 1 • Therefrom,
n
U0 n R(T*) = U0
T* N°
is a(Et,E)-closed in E' and, by the Krein-Smulian theorem, R(T*) is a(E•,E)-closed, hence the result, by prop.VI.1.10. PROPOSITION VI.2.4. linear map of E
~
~
E,F
~ ~-spaces
and T a continuous
F. The following conditions are equi-
valent : (a) T is surjective, (b) for every neighborhood U of zero in E, there exists a vector subspace N R(T) + N
~
F such that
= F
T*- 1 U° C N° •
(c) there exists a non increasing sequence Nk(k E-lN) of closed subspaces ofF such that R(T) + Nk
= F for each k
00
n
Nk =
k=1
{o}
and that, for every neighborhood U of zero in E, T ~
a
=>
-1
Nk C ker T + U
k large enough. b and c
TakeN= Nk = {0},
112
e-m,~
b
)
a
It is proved in prop.VI.2.3. c
=>
a
Let Uk(k E-lN) be a base of neighborhoods of zero in E such that 2Uk+ 1 C Uk for each k E- lN. By a suitable relabelling, we may assume that T- 1 Nk C ker T + Uk and R(T) + Nk
=
F, Ilk E-lN •
Given y E- F, for each k E-m, there exists yk E- Nk such that y - yk E- R(T). Hence yk- yk+1 = (y-yk+1) - (y-yk) E- R(T) n Nk C TUk • OCI
Choose xk E- Uk such that Txk = yk+ 1 • The series
E xk is conk=1
vergent to some x E- E. Moreover, k
k y
-
T
E xi i=1
Y- .E (yi-yi+1) l.=1 E- y - y 1 + Nj,
Ilk~
=
y - Y1 + yk+1
j
•
Since each N. is closed. J
00
y 1 - Tx
= (y-Tx) - (y-y 1 )
Thus y 1 E- R(T) and y
E-
n j =1
N ... J
{o I
•
y-y 1 + y 1 E- R(T). Hence T is onto.
3. LIFTING COMPACT SUBSETS We have seen in prop.VI.1.5 that a linear map T is almost open if and only if every equicontinuous subset of R(T*) can be lifted by T* to an equicontinuous subset of F•. If T is a weak homomorphism, R(T*) is cr(E 1 ,E)-closed. It is enough to lift equicontinuous sets of the form
uo n
R(T*), where U is a neigh-
borhood of zero in E; these are absolutely convex and cr(E 1 ,E)compact. If F is a Mackey space, every absolutely convex and compact subset of
F~(F•,F)
is equicontinuous. So the question 113
whether a relatively weakly open map is relatively open turns out to be closely related to the possibility of lifting weakly compact sets by its adjoint. This has already been used in prop.VI.1.9,b) where the lifting problem was made obvious since T* was injective, and in prop.VI.1.10,f). Let us first investigate how, conversely, the lifting problem translates by duality. DEFINITION VI.).1. LetT be a linear map of E into F and$ (resp. $
1 )
be a collection of subsets of E (resp. F). We say
that T lifts $ such that Te
to$ if, for every e' f $
1
e
1
1 ,
there exists e f $
n R(T).
It is often sufficient to find e such that Te ) e 1 1 it suffices that en T- 1 (e 1 ) f $ i f e f $.To check this, the following lemma will be useful. LEMMA VI.).2. ~ T be a linear map of E ~F. closed in E x F, for every compact subset K
s£
li. G(T) l l
F, T
-1
K is clo-
sed in E. li. G(T) is fast sequentially closed and K fast compact, T- 1 K is fast sequentially closed in E. Let x (afU) be a generalized sequence in T- 1 K, converging 0:
to x. Since K is compact, Tx (afU) has an adherent point y f K. 0:
It is clear that (x,y) f
GTTJ =
G(T), hence Tx = y and xf T- 1 K.
For the second case, replace x (o:fU) by a fast convergent 0:
sequence xn(nflN). Then Txn(nflN) admits a fast convergent subsequence and the conclusion follows as above. PROPOSITION VI.).). ~
~
T be a linear map of E .i.!:!..:tQ F ..!D!.Q.h
G(T) is closed in E x F ~ $ ~ $' be covering families
of bounded subsets of E
~
the two following assertions (a) T lifts $
1
F, both contained in $ • Consider
•
~ $,
(b) T* is relatively open of ~(T*) equipped with the topology t$ 1 .i.!l.i£ E~.
114
~ b
~
b => a and, 1L R(T) is closed, a
~
b.
a
If b is true, given K1 T* K'
0
~
there exists
~·,
Kin~
such that
K0 n R(T*) '
)
or T*- 1 K° C K10 + ker T* • Take the polar of both sides {K 10 + ker T*)° C {T*- 1 K0
) 0
•
Observe that K1
n
RtTJ
C (K
Indeed, if x ~ K•
+ ker T*)
10
1l
iiTT') and y 1
0
•
f
K'
y2 f
0 ,
ker T* ,
Thus
Now,
(T*- 1 K0
) 0
n R(T)
Il ~ 1, ~y f T*- 1 K0
{Tx .. {Tx:
l<x,T*y>l ~ 1, ~y E- T*- 1 K 0
T(K 0 nR(T*)) 0 Since~
C
~~
}
•
fort~
and for o(E 1 ,E) are the same. But K0 is an absolutely convex and closed neighborhood of zero for t~. Thus 0 -n~R~(~T~*T) ~K~
, the closure of R(T*)
}
t~
= K0
n R(T*)
t~
(cf. the proof of (*) in prop.VI.1.5). This equality reads
___.,..........,.o(E•,E) K0 nR(T*}
= K0
li
(ker T) 0
•
Thus K + ker T •
11 5
For the last equality, observe that (K 0 n(ker T) 0 ) 0 = 'Cc;(KUker T)
=
U [eK+(1-9)ker T] eE-[o,1]
=K+
ker T
since K is compact and ker T closed. So, finally, K' n R(T) C T(K+ker T) = TK • a ~ b (when R(T) is closed). Given K 1 Eo~·, there exists K
Eo~
such that
K I n R ( T) C TK , or T
-1
K 1 C K + ker T •
Therefrom, (K+ker T)° C (T- 1 K•) 0 But K0 n R(T*) C (K+ker T) 0
•
o
Thus
K0 n R(T*) C (T- 1 K•) 0 n R(T*) • We have {T*y: yE-f>(T*) and l<x,T*y>l~1, llxE-T- 1 K 1
= T*(K 1 nR(T)) 0
}
•
Since R(T) is closed, (K'nR(T)) 0
K1° + R{T) 0
=
K1° + ker T*
o
Thus K0 n R(T*) C T* K' 0 and T* is relatively open as announced. We care now for lifting properties of compact sets. A first classical and easy result is the following. PROPOSITION VI.}.4o ~ E,F be metrizable spaces and T an open map of E onto F. 1..h.!Jl, every convergent sequence of F is the image by T of a convergent sequence of E. Moreover, 1L E is an ~-space and if ker T is closed, every compact subset of F iA the image by T of a compact subset of E.
11 6
There exist bases of neighborhoods of zero in E and F, Un (n E-lN) and Vn (n E-lN) such that TU n ) Vn , Un+ 1 C Un and Vn+ 1 C Vn for each n E-lN. Let yi(iE-lN) be a convergent sequence of F. We may assume that yi such that y i
<
for in~ i proof.
E- Vn for i
~
o.
=
There exist then in t
in. Choose xi E- Un such that Txi = y i
in+ 1 • It is clear that x i - 0 in E, hence the
If K is compact in F, there exists a sequence ym -
0 such
that K C Cc;'({ym am E-lN}) (cf.prop.III.1.4). Let x is an 9-"-space, K'
m
- 0 in E be such that Tx
m
c
y • If E m
: m E- lN}) is compact in E. m Since T is open and onto and ker T closed, the induced map
T of
Cc;'( {x
c
E/ker T onto F has a continuous inverse, hence G(T) is
closed in E/ker T x F and G(T) is closed in E x F. By prop. III.1.4, each y E- K can be written y =
=r:
c
m=1
=r:
with
m=1
I cml
m ym ~
1
0
If
00
X
=
r: m=1
c
m
X
m
since G(T) is closed, we have Tx = y, thus TK'
) Ko
Therefore, K = T(K•nT- 1 K), where Kt n T- 1 K is compact, by lemma VIo3o2o Prop.VIo3o4 is considerably improved by the next result, due to Fakhoury
[36].
PROPOSITION VIo3o5o
~
E,F be l.c.s. and T a linear map of E
into F. lL T is open and if ker T
~'
as a subspace of E,
9-"-space, then every compact subset of F is the image by T
~ ~
compact subset of E.
117
This result can actually be improved by constructing a continuous (non linear) right inverse ofT
[36].
The following simple proof of prop.VI.3.5 is due to
J. Zafarani {unpublished). We need a lemma. ~
LEMMA VI.3.6.
E,F be l.c.s., K a compact subset of E
a map of K into the set of all convex subsets of F,
~
{x ~ K: ~(x)
n
~
~
-,'¢} l l open in K. ~' for every absolutely convex neighborhood V .!2£ zero in F, there exists a continuous map ~ .!2£ K in!s F ~ that, for every open subset w .!2£ F,
w
.llull ~(x)
~ ~(x) + V,
~x ~ K •
We call ~ a V-selection of ~· We may assume that V is open. The subsets U ={x~K:y~tp{x)+V} y
= {x~K:~(x) n {y+V)
1¢}
(y~F)
determine an open covering of K. There exist thus a finite covering U (i~n) of K and a continuous partition of unity of K, yi ai(i~n), such that supp ai C U for each i. Define yi n
w(x) =
E ai(x)yi • i=1
If ai(x) ~ O, we have yi ~ ~(x) + V. Moreover ai(x) ~ 0 and n
a.(x) = 1. Thus ~(x) ~ ~(x) + V for each x ~ K.
E
i=1
l.
Proof of prop.VI.3.5. Let K be a compact subset of F and let Un (n
~lN)
be a sequence of absolutely convex neighborhoods of
zero in E such that U
n
n
ker T (n ~lN) is a base of neighbor-
hoods of zero in ker T and that 2Un+ 1 C Un for each n ~lN. By lemma VI.3.6, there exists a u1 -selection ~ 1 of the map
~1
: y - T- 1 y.
Indeed,
{y ~ F
T- 1 y n w -,'p} = Tw is open in
F whenever w is open in E. We determine then by induction a sequence of maps ~ n (n ~lN) such that ~ n+ 1 is an Un+ 1 -selection of 11 8
Each
verifies the assumptions of the lemma. Indeed, given
~n
an open subset w of E, consider
If y 0
o,
~
there exists x 0
~
= y0
E such that Tx 0
,
x0
~
w and
x 0 ~ ~n(y 0 ) + Un. Choose an absolutely convex neighborhood U of zero in E such that x 0 + U C w and x o - q, n (y o ) + 2U C Un and a neighborhood V of zero in F such that TU ) V and q,n(y) - q,n(y 0
)
~ U when y ~ K andy - y 0
f
v.
Then, if y ~ K
and y - y 0 f V, there exists x such that Tx Thus
X
f
w
and
X-"'
n (y) ~
X
o
-"'
y and x- x 0
U.
~
n (y o ) + 2U ( un. Hence g is
open in K. For each y ~ K, ~j.n(y) (n E-lN) is a Cauchy sequence. Indeed, "'n+1(y) E- ~n(y) + un+1 ( ~~>n(y) + un + un+1 thus lj.n+ 1 (y) - q,n(y) f
2Un for each n ~ 2. Moreover, all the
q,n(y)'s belong to T-1y, which is a translate of the U-space ker T, for which U (n fiN) is a base of neighborhoods of zero. n 1 Thus the sequence ~j.n(y) (n fiN) converges to ~j.(y) f T- Y• The map q, is continuous in K. Indeed, q,n converges to
ljJ
uniformly on K s-1
s-1
~
~
i-r
i=r
Thus
Since Toq,(K)
ljJ
=
is cor.tinuous, q,(K) is compact in E. Moreover
K, hence the proof.
REMARK. The convexity of K is not preserved by the lifting, so that prop. VI.}.5 does not provide information on the lifting of
~ca
to
~ca"
Moreover, prop.VI.}.5 can hardly be used
for weak compact sets. Indeed, the assumption on ker T would 119
be that it is an ~-space for the topology induced by o(E,E•), which is the weak topology of ker T. But, if a l.c.s. E is an ~-space for the topology o(E,E 1 ) , it is isomorphic to lRw with w finite or countable. Indeed, if y n (n E-lN) E- E 1 are such that
{yi: i~n} 0 (nE-lN) is abase of neighborhoods of zero in E, it is clear that E 1 is the linear hull of {y n : n E-lN}; so its dimension is countable. Let {yi ~ i E- w) (w ClN) be a basis of E'. Every element x of E is characterized by the sequence <x,y.>(i E- w) and J : x <x,y.>(i E- w) is an isomorphism of E 1 1 intomw. The map J is surjective. Indeed, for each sequence c. (i E- w), there exists a unique linear form x on E' such that 1
Jx = c. (i E- w). Since the equicontinuous subsets of E 1 are fi1
nite-dimensional, x is continuous on each equicontinuous subset of E' for the topology o(E 1 ,E). By the Grothendieck 1 s completion theorem, x is thus an element of E. Hence the result. There is another interesting lifting theorem for webbed spaces. PROPOSITION VI.}.7. ~ T be a linear map of a subspace ~(T) ~
E onto F. If the graph of T is fast sequentially closed and
E strictly webbed. a) every fast convergent sequence of F is the image by T ~ fast convergent sequence of E, b) every fast compact subset of F is the image by T of a fast compact subset of E. a) Let ym(m E-lN) be fast convergent. It is not a restriction to assume that its limit is zero. We prove that there exists a fast convergent sequence x
m
-
0 such that Tx
m
y • m
Let B be a bounded Banach disk of F such that ym- 0 in FB. Consider the linear relation yRx <=> Tx .. y of FB into E. Its graph is fast sequentially closed since, up to the order of the factors, it is G(T) n (ExFB). By prop. IV.6.1 (the localization property), there exists a strand 120
e
(k E-lN) of a strict web fli of E and numbers Ck \lk that
>
0 (k E-lN) such
In other words, Ck T e
\lk
) B, Vk E- IN •
Let t..k(k E-lN) be strictly associated to the strand e
(k E-lN) \lk (we may assume that t..k ) 0 by taking a subsequence) and
e:k
>
(k E-lN) be such that
0
e:k B
<
t..k T e
\lk
, Vk E- IN •
Since ym - 0 in FB' there exist mk t all m ;:, mk. For mk
~
~
such that ym E- e:kB for
m ( mk+ 1 , choose xm E- t..k e\lk such that T xm
= ym'
0 in E. The sequence x (m E-lN) is even It is clear that xm m fast convergent, for a suitable choice of t..k(k E-IN). Indeed, take t..k
= pk t..k' where t..k(k E-IN) is another strictly associated
.
-1 sequence and pk J. o. I f x' pk X m (mk ~ m m x'm ~ t..'k e for mk ~ m < mk+ 1 • Each series \lk 00
<
mk+ 1)' we have
00
em x'm ( .E m.. 1 m=1 .E
19 m I
~
1)
is thus convergent. Indeed,
~
A.'k
thus 00
E
k=1
em
x') m
is convergent and
121
x' E- >..'k m for mk
~
r
<
mk+ 1 , hence converges to zero if k -
~.
By prop.III.1.4, 00
K = { E m=1
00
em
x'm
E
m=1
is thus a bounded Banach disk of E and xm E- pkK (mk~m(mk+ 1 ) hence xm -
0 in EK.
b) Let now K' be fast compact in F: K 1 c"Cc;'({ym: mE-JN}), where y
m
is fast convergent to zero in F, by prop.III.1.5. Let
xm(m E-lN) be the lifting of ym(m E-JN) considered in a). It is fast convergent to zero, hence K = "Cc;'({x
m
: m E-lN})
is fast compact in E. We have TK ) K 1 •
Indeed, if y E- K•, for 00
~
sui table ci ( i E-lN) such that
E
E
i=1
i=1
The series is fast convergent (indeed, "Cc;'({y
m
:mE- lN}) is com-
pact in some Banach space FB 1 and the series is convergent to 00
y in FB 1 ) . Define x 00
the same argument,
E c 1 xi. It is an element of K and, by i=1
E ci xi is fast convergent to x. Since i=1
G(T) is fast sequentially closed, it follows that Tx = y, hence y E- TK. Assume finally that K is compact in the Banach space FB , that K' is compact in the Banach space FB' and that TK ) K 1
•
Then T K 1 =T(K n T- 1 K'), where T- 1 K 1 is closed in EB, by lemma VI.}.2. Hence the result. We apply now prop.VI.}o7 to the study of homomorphisms.
122
PROPOSITION VI.).8.
Let T be a continuous linear map of E ln!2
F. Assume that E is a Schwartz space, and that F is evaluable
Fb
~
webbed.
a) T is almost open if R(T*) is fast sequentially closed in E~(E•,E)"
The converse is true when E i s a Mackey space.
b) T is relatively open if and only if R(T*) is closed in E~(E',E)'
or if and only if Tis relatively weakly open.
a) Let U be a neighborhood of zero in E. Since E is a Schwartz space, there exists another neighborhood U 1 of zero such that U0 is compact in E E~(E•,E)" Since R(T*)
0,
0 ,
hence fast compact in
is fast sequentially closed,
uo n
R(T*)
is again fast compact for o(E ,E). 1
Fb
Since Tis continuous, T* is continuous of into E~(E•,E)" Using prop.VI.3.7, there exists a compact disk K of such that T*K )
uo n
Fb
R(T*). But F is evaluable, so that K is equi-
continuous. r,onclude by prop.VI.1.5. Conversely, if T is almost open, for every neighborhood U of zero in E, there exists a neighborhood V of zero in F such that R(T*)
n U° C
T* V 0
R(T*)
n
U0
•
Thus U0
=
n
T* V 0
is o(E•,E)-closed in E•. Therefrom, R(T*) n E the Banach space
Euo•
0
o
is closed in
Let now x~(m~m) be contained in R(T*)
and fast convergent to x
in E'. By prop.III.1.7, there exists s a compact disk K' of E~(E•,E) such that x~- x 1 in EK'" Since E is a Mackey space, K' is equicontinuous. If K 1 C U 0 , it is clear that x~- x' in E
1
00 ,
hence x' ~ R(T*).
b) Since T is continuous, it is relatively open if and only if it is relatively weakly open and almost open (prop.VI.1.6) and it is relatively weakly open if and only if R(T*) is o(E•,E)-closed (prop.VI.1.4). Hence the result, by a).
123
PROPOSITION
VI.~.9.
~
E be the inductive limit of a sequence
of metrizable Schwartz spaces Ei (i E
~
t be the topology of E
t
~
1
the topology of the
inductive limit of the spaces L n E.(i E-lN). ~
a) The topologies t ~ t' coincide on L if and only if ( Lt ) I b)
1£
=
( Lt
I ) I •
E i s a strict inductive limit, and if (Lt)
the canonical identification of Eb/L
0
1
=
(Lt
1 ) 1 ,
~ (Lt)b is an isomor-
phism and every bounded subset of (Lt)b can be lifted to a bounded subset of Eb. a) It is clear that t ~ t' in L. Thus the identical map J of Lt 1 into E is continuous. Moreover, E is evaluable, its dual is webbed and L is a Schwartz space (since each L n Ei is a Schwartz space). We can thus apply prop.VI.~.a : if (Lt)' = (Lt
1 ) 1 ,
we have J*E'
=
L1
,
hence J is relatively open.
Being injective, it is thus an isomorphism of Lt b) If the inductive limit E is strict, Lt inductive limit. By
prop.III.3.1~,
each L
n
1
into Lt.
is also a strict
1
E. has an ultrabor~
nological strong dual. By prop.III.3.15, (Lt,)b is thus ultrabornological. Moreover, every bounded subset of Lt' is contained in the closure of a bounded subset of some L n Ei. We have thus (Lt 1 )b = (Lt)b. The canonical map n : Eb- (Lt)b is continuous and surjective, the range space is ultrabornological and the domain space is webbed. By prop.IV.5.2,n is thus open, hence Eb/L 0
is
isomorphic to Lb. Moreover, every bounded subset B of Lb is equicontinuous. Since L is a Schwartz space, it is contained in a fast compact disk K of Lb. By prop.VI.~.7, K can be lifted to a fast compact disk of Eb, hence to an equicontinuous subset of El. REMARK. The subspac~ L of E such that (Lt)' = (Lt 1 ) ' are called well located. They are studied in [40] and [100-102].
124
PROPOSITION VI.3.10.
~ T be a continuous linear map of E F. 1! E is strictly webbed and F the strict inductive E-IN), the following three limit of a sequence of ~-spaces F.(i 1 conditions are equivalent
~
a) R(T) is closed in F, 1 b ) T* -.,.i.:;s=.....:r~e:::..:.l.iiila...=t~i:..:V:..:e:..l:.Y.t-..l:oup~e::.:n~~o-=-f F c1 ~ E ca
c) T* is relatively open of F~ ~ E!• Since the inductive limit is strict, every compact subset ofF is contained and compact in some Fi. It is thus fast compact. Therefore ~ c (F) = ~ ca (F) C ~ -r (F). It follows that (F')' = F and (E'ca ) 1 =E. By prop.VI.1.4, a) is thus equivac lent to c) and, by prop.VI.1.6, b) ~c). Since every element of ~ (F) is fast compact, by prop. c VI.3.7, T lifts ~c(F) to ~ca(E). Thus, by prop.VI.3.3, a) ~ b).
125
VII Souslin spaces and the Borel graph theorem ofL. Schwartz 1. SOUSLIN SPACES DEFINITION VII.1.1. A Souslin space is a regular Hausdorff topological space X such that there exists a continuous map of a separable and complete metric space onto X. Denote by I
the set
{(n 1 , ••• ,nk) : k, n 1 , ••• ,nk f lN} ordered by
Let X be a topological space. A seave
~
of X is a map v -
e
v
of I into the set of all non empty subsets of X such that (a)
e~
(b) 11
~
x,
=
v
l> e
e
)
11
v
, ,
u ev 1111 f I v>J.l (d) for every sequence vk(k E-lN) such that vk < vk+ 1 for all k f lN, the sequence e (k flN) is a base of a convergent filter vk of X. (c) e
1.!
PROPOSITION VII.1.2. A regular Hausdorff topological space X is a Souslin space if and only if it admits a seave. Assume first that X is a Souslin space : it is the image by a continuous map
~
of a separable and complete metric space Y.
Let d be the distance of Y and {y.l. : i f subset of Y. Define k
e(n,, ••• ,nk) =
~(i~1
{y : d(y,yn.)
It is obviously a seave of X. 126
l.
m}
a countable dense
< tJ)
Conversely, assume that X admits a seave The space I
lN
~.
, where each factor I is equipped with the
discrete topology, is a separable, complete metric space. Denote by I
0
the subspace of IlN,
It is still a separable, complete metric space (equipped with the distance induced by IlN ). Consider the map ~p(vk(k
e-:nn>
The map
~
of I
0
into X defined as follows
:
is the limit of the filter generated by evk (k E-lN). 1p
is continuous. Indeed, let U be a neighborhood of
x "' ~(vk(k E-lN)). Since the filter generated by ev (k E-lN) is C U for k ~ k 0
convergent to x, we have e
vk By definition of the topology of IlN, I
0
:
I.L
Vk :IE k
k =
0
k
(k 0 large enough).
)
is a neighborhood of vk(k E-lN). For all I.Lk(k E-lN) E- V, we have e ~v
C U for all k
vk
~
k0
;
thus, if U is closed,
0
c u. The map
1p
is onto. Indeed, by (a) and (c) in def.VII.1.1,
for each x E- X, there exists a sequence vk(k E-lN) in I such that vk
< vk+ 1
and x E- ovk for each k E-lN. Then x =lim evk =
Comment. A seave is a web satisfying the additional condition (d). The concept of web was actually suggested by the seaves, taking advantage of the vector space structure to relax condition (d) in the definition of completing webs. The following proposition relates Souslin spaces to webbed spaces. PROPOSITION VII.1.3. A fast complete Souslin l.c.s. is webbed. Let~=
{e
v
: v E- I) be a seave of the l.c.s. E. It is a
(k E-JN), the filter vk generated by this strand is convergent. Thus, if xk E- e (k E-lN), vk completing web. Indeed, for each strand e
127
the sequence xk(k E-lN) is convergent, hence bounded. The conclusion follows by prop.IV.1.8. 2. PERMANENCE PROPERTIES AND EXAMPLES PROPOSITION VII.2.1. Let X be a regular Hausdorff topological space and X (m E-lN) be subspaces of X. If each X is a Souslin m m 00
space, ..i!':uu!
..
I f 'lfm
u m=1 {e(m) v
Xm is a Souslin SJ2ace. I
v E- I) is a seave of Xm for each m E- lN, 00
then (i,v)
-o
e (i) is a seave of u Xm• v m=1
PROPOSITION VII.2.2. The product of a sequence of Souslin spaces is a Souslin space. Let X (m E-lN) be Souslin spaces. If each X is the image by m m a continuous map ~m of a separable complete metric space Xm , 00
n
then
X
m
m=1 ~(y
m
is the image by the map
(mE-lN)) ..
1p
m
(y m )(mE-lN)
00
of
n Xm , which is still a separable, complete metric space. m•1 Also, if 'If = {e(m) : v E- I) is a seave of Xm for each m, m
v
the subsets 00
e(m) vm • m=1
n
where all but a finite number of the v 00
seave of
n
m=1
m
1
s are equal to
p,
is a
X (see the corresponding property of webs, prop. m
IV.4.8). PROPOSITION VII.2.3. A continuous image of a Souslin space is a Souslin space. Straightforward consequence of the definition.
128
PROPOSITION VII.2.4. A Borel subset of a Souslin space is a Souslin space. Let X be a Souslin space. Every closed or open subset of X is still a Souslin space. Indeed, if X is the continuous image by !p of a complete separable metric space Y, and if e c X is closed or open, then ll'-1(e) is still C 1 OS ed or open in Y, hence it is another separable complete metric space. We could as well use the characterization by seaves. If is a seave of X, is closed and {e
{e
n e =I=¢;
v
: v E- I and e
v
v E- I}
v
~
is a seave of e when e
C e} is a seave of e when e
is open. Every countable intersection of Souslin subsets e (mE-IN) of m
co
n e is isomorphic to m=1 m , thus to a closed subspace of a Sous-
X is again a Souslin space. Indeed, co
the diagonal of
rr
m=1
e
m
lin space. The above results and prop.VII.2.1 show that the class of subsets e of X such that e and Ce are Souslin spaces is stable under countable unions and intersections. It is moreover stable under the map e -
Ce and contains the closed subsets of X.
Hence it contains all Borel subsets of X. COROLLARY VII.2.5. Every countable projective limit of Souslin spaces is a Souslin space. Indeed, it is isomorphic to a closed subspace of a countable product of Souslin spaces. REMARK. There is a slight improvment of prop.VII.2.3 and prop. VII.2o4• Indeed, if X is the image by
q~
of a Souslin space Y
and if IP is sequentially continuous, X is a Souslin space. For, if Y is the image by ~ of a separable complete metric space z, then X
=
~po~Z
and
q~o~
is sequentially continuous, hence conti-
nuous. Also, if e is a sequentially closed subset of a Souslin space X, e and Ce are Souslin space 1 if X • q~(Y) (q~ continuous,
129
Y separable complete metric space), e = ~ ( ~ sequentially closed hence closed in Y,
_, e ) ,
when ~
and Ce = ~(c~- 1
_, e
is
e).
DEFINITION VII.2.6. Define the set of sq-Borel subsets of X as the smallest class of subsets of X, stable under the map e -
ce, under countable unions and under countable intersec-
tions and containing the sequentially closed subsets of X. PROPOSITION VII.2.7. A sg-Borel subset of a Souslin space is a Souslin space. Combine the proof of prop.VII.2.4 with the above remark. PROPOSITION VII.2.8. The image of a Souslin space by a sequentially continuous map is a Souslin space. See the above remark. There are interesting examples of Souslin l.c.s. PROPOSITION VII.2.9. Separable
~-spaces
and countable induc-
tive limits of such spaces are Souslin spaces. For separable
~-spaces,
it is straightforward. A countable
inductive limit of such spaces is a countable union of continuous images of separable VII.2.1 and 3.
~-spaces,
hence the result, by prop.
PROPOSITION VII.2.10. The dual E 1 of a separable and metrizapc ble space E or of a countable inductive limit of such spaces is a Souslin space. Let E be metrizable and separable and let U (m ~m) be a base m of neighborhoods of zero in E. Then each U0 , equipped with the m
topology induced by E~c' is separable, metrizable and complete [43, pp.227-22B or 51, p.235-252]. Since E•
= ""U
m=1
U0
,
the conclusion follows from prop.VII.2.1.
m
Let now E be the inductive limit of metrizable and separable spaces Ei ( i ~m). The dual of E can be identified to the L=
130
fxl(i~m):
IIi ~
m}
subspa~
co
of
n
Ej_.
i=1
co
n
By prop.VII.2.2,
i=1
00
closed in
n i=1
E! is a Souslin space. Since L is l.,pc
E!l.,pc it is thus a Souslin space for the induced
topology. It is however not isomorphic to E'pc , since every precompact subset of E does not need to be a precompact subset of some Ei. We will show that the canonical map of L into
E~c
is sequentially continuous and conclude by prop.VII.2.8. It is equivalent to prove that the identical map of E 1 into itself is sequentially continuous of
into E 1 "., pc the set of all precompact subsets of all Ei. E~
where .7J is
,
Let x' (m ~m) be convergent to zero in E~. The sequence m x' IE (m ~IN) is bounded in E! , thus equicontinuous in Ei• m i l.,pc Hence x' (m ~lN) is equicontinuous in E•. Since every equicontim
nuous subset A of E 1 is relatively compact in E'pc , the topolocoincide in A. Hence x'm 0 in E'pc PROPOSITION VII.2.11.
~
E be metrizable and
separable,~
countable inductive limit of such spaces, and let F be a separable ~-space. ~ L (E,F) is a Souslin space. pc Assume first that E is metrizable and separable. Denote by pi (i ~m) (resp.qi (i ~m)) a sequence of continuous semi-norms of E (resp. F) characterizing its topology. We may assume that
{x ~ E : pi (x) ~ 1 I (i ~IN) is a base of neighborhoods of zero in E. Denote by {x.l. : i ~ lNI a countable dense subset of E and by e ( vH) a seave ~ of F. Define v
e
( i) v
{T ~ L(E,F)
Tx. ~ e I l. v
and
• We define a seave ~· of L(E,F) as follows non empty subsets of the form
we consider the
1 31
n ••• n
e ( k) n u< 1 ) n ••• n vk n1
u ( k) nk
and the order relation
k
<
k
mi
1 ;
By a suitable relabelling, this takes the usual form of a seave. It is trivially a web. To check condition (d) of def. (k~IN)
VII.1.1, observe that a strand & sequence xk(k
~lN)
such that, if Tk qi(Tkx) ~ p and Tk xi
~
of
j!k
ni
determines a
1
.
and, for each i, a strand ~
'ft
v~ 1 )(k ~lN)
of
'ft
& , j!k
(x), vx, Vi~ k , ~
e (i) , Vi vk
(*) (**)
k •
The sequence Tk(k~lN) is equicontinuous by (*). Moreover, for each i ~IN, Tk xi is convergent in F by(**). By the Banach Banach-Steinhaus theorem, the sequence Tk is thus convergent in Lpc (E,F) to the map T defined by Txi
= lim Tk xi = klim ev(i) • k-oeo
......,.,
k
This limit is independant of the choice of Tk
~
& , thus "'k
Let now E be the inductive limit of the separable and metrizable spaces Ei (i ~lN). The spac.e L(E,F) is isomorphic to the subspace L
=
{Ti(i ~IN): T.
1+
1 IE i
=
T., Vi ~lN} 1
00
n L(E.,F). Each Lpc (E.,F) is a Soualin space and Lis 1 1 i=1 00 closed in n L (E. ,F). So L(E,F) is a Souslin space for the i=1 pc 1 of
corresponding topology, i.e. the topology 1 32
t~,
where
~
is the
set of precompact subsets of the E. 's(i E-JN). 1
The identical map of L~(E,F) into L (E,F) is sequentially oA> pc continuous (same argument as in prop.VII.2.10, using the Banach-Steinhaus theorem), hence the conclusion, by prop. VII.2.8.
3. SOME PROPERTIES OF MEAGRE SETS DEFINITION VII.3.1. Let X be a topological space and e a subset of X. We denote by D(e) the set of all x E- X such that, for every neighborhood V of x, V
n
e is non meagre. We denote
by O(e) the interior of D(e). In other words, O(e) is the set of points x such that there exists an open neighborhood w of x such that, for every non empty open subset w' of w, w' n e is non meagre. ~ w
PROPOSITION VII.3.2.
a
(aE-Q) be a collection of pairwise
disjoint open subsets of X. l! e C UaE-Q wk gre for each a E- Q, ~ e is meagre.
~
e
n
wa is mea-
For each a E- Q, there exists a sequence F n,a of closed sub0
sets such that F w
•
n,a
p
and e
n
w
c
a
u
F
n=1
n,a
• Put
n F n,a • It is clear that e C u e n • For each n, n=1
a 0
u w
aE-Q a
U
U
aE-Q
(w nF n a
,a
)
0
0
If en has an interior point x, since
U w has a void interior, aE-Q a 0
x is an interior point of some w 0
n,a. Thus en
=
p and
n F n,a • But F n,a • a
p for
all
e is meagre.
PROPOSITION VII.3.3. For every subset e ~X, e \ O(e) iA meagre. It is c 1 ear that 0 [ e \ 0 ( e) ]
p.
Indeed,
133
e' C e ~ O(e') C O(e) implies that O[e \O(e)] C O(e), and O(e
C
1 )
et
D(e 1 ) C
implies that O[e \O(e)] C e \ O(e) C CO(e). ~.
So it is enough to prove that e is meagre if O(e) note by
e
the set of all collections {w
a
disjoint open subsets of X such that w a f
u.
a
e
Order
a f
1
u}
De-
of pairwise
n e is meagre for each
e
by inclusion. Every totally ordered subset of
has an upper bound, the union of its elements. By Zorn's lemma, there exists a maximal element
U
w
a a f"' u 0
{w
= X. Indeed, since O(e) =
a ~'
: a f
fJ } of
o
e.
We have
X and every
for every x f
neighborhood V of x, there exists an open subset w of V such that w n e is meagre. If x
u
f
, choose V disjoint from
w • Then {w} U {w a : a f
large than {w
a a fn u 0
a
1
a f
fJ },
o
hence a contradiction. We have thus
u
w ) a f"' u 0 a
e C ( U
•
where ( U w ) afU 0 a well as
u
(enw ) ,
aff/ 0
a
is meagre (because it has a void interior), as
(enw ), by prop.VII.3.2. Hence the result.
U
a
afU 0
PROPOSITION VII-3.4. 1£tl en(n fiN) be a sequence of subsets of 00
00
x. l!.h.!.!'! o( u
e )\ U O(e ) is meagre. n n=1 n n=1 00
Since [ U noo1
has a void interior, it is sufficient to 00
prove that o( u n=1
00
u n=1
00
If x f
0(
U
n=1
00
en), for every neighborhood V of x, V n
U
n=1
e
n
is non meagre. By prop.VII.3.3, each en \O(en) is meagre. Howe-
1 34
00
00
v n
u en n=1
c
CIO
u en\ O(en) u [v 11 u 0 (en)] • n ... 1 n=1
00
Thus v n
u O(en) is non meagre, hence non empty. n=1
PROPOSITION VII.3.5.
~
X be a topological group and a Baire
space. lL O(e)\ e is meagre and O(e) ~¢, ~ O(e).o(e)- 1 C e.e- 1 • Take x ~ O(e).o(e)- 1 • Then O(e) n xO(e) ~¢. It is open, hence non meagre. Setting 0( e)\ e = e 1 , we have O(e) n xO(e)
c
(eue 1
)
n x[eUe 1 ] C (enxe) u e"
where e" is meagre (because e' is meagre). Thus e n xe is non meagre, hence non empty and x ~ e.e -1 • PROPOSITION VII.3.6. [72]. ~X be a l.c.s. !£ e C X is absolutely convex, non meagre and such that O(e) \ e is meagre, ~ e is a neighborhood of zero. Since e is non meagre, by prop.VII.).3, O(e) is non empty and even non meagre. Let us prove that O(e) C e-e. It follows that e = f<e-e) has interior points, hence that it is a neighborhood of zero. Given x ~ O(e), there exists A~ ]0,1[ such that x ~ AO(e) (because O(e) is open). Thus O(e) n [o(e)-x] ) (1-A) o(e) - x • The right-hand side is non meagre, hence O(e) n (O(e)-x) is non meagre. If e' = 0( e) \ e, we have O(e) n [o(e)-x]
c
[en(e-x)] u {e•n[o(e)-x]} u [o(e)n(e 1 -x)] •
The set [e 1 n(o(e)-x)] u [o(e)n(e 1 -x)] is meagre, hence en (e-x) is non meagre, thus non empty, and x ~ e-e.
135
PROPOSITION
1L
space X.
VII.~.7.
~
e be a subset of a regular Hausdorff
e is a Souslin space, O(e)\ e is meagre.
Let e (v~I) be a seave of e. By prop.VII.~.4, e =
v
U
v>P
e
v
implies that
flp
o( e) \
=
u
O(e ) v
v>P
is meagre, and, for each g ~
0( e ) \
=
~
u
~'
e
e
I!
v
implies that
O(e ) v
v>I.L
is meagre. Let us prove that O(e)
(
e
u
u v~I
g
v
This shows that O(e)\ e is meagre. Given x ~O(e)\
U v~I
there exists v 1 such that x ~ O(e e
vk
x E
(k ~lN) such that x ~ O(e
k of e.
n e ~lN
E
vk
vk
v, )
g
v
and by induction a strand
) for each k ~lN. Thus
e, since the filter e
vk
converges to some point
COROLLARY VII.~.e. ~X be a l.c.s. Every absolutely convex, non meagre and Souslin subset of X is a neighborhood of zero. Let e be such a subset. By prop.VII.~.7, O(e) \ e is meagre. The conclusion follows from
prop.VII.~.6.
4. THE BOREL GRAPH THEOREM FOR L.C.S. We prove now the Borel graph theorem of L. Schwartz. The present proof is due to Martineau [72]. PROPOSITION VII.4.1.
~
E be ultrabornological, Fa Souslin
l.c.s. and T a linear map of E ~ F such that G(T) is a sgBorel subset of E x F. ~ T is continuous.
1 ~6
Assume that E is the inductive limit of the Banach spaces E (a~v). Consider the restriction T of T to E • Its graph a a a G(T) n (E xF) is a sq-Borel subset of E x F. Indeed, the set a
a
of all subsets e of E x F such that e
n
(e xF) is a sq-Borel a x F contains the sequentially closed subsets of
subset of E
a E x F and is stable under countable unions and intersections
and under complements. Thus it contains the sq-Borel subsets of E x F. If the result is proved for E (a~V), each T is continuous a a and T is thus continuous. So we may assume that E is a Banach space. We may even assume, when E is a Banach space, that it is separable. Indeed, T is continuous of E into F if it is sequentially continuous. If x (m E-lN) is a sequence of E, its closed linear hull is a sepam rable Banach space E 0 and it is enough to prove that T is continuous on E 0
•
Now, by prop.VII.2.9, a separable Banach space is a Souslin space. It i3 moreover a Baire space and we have left to prove the result when E is a Baire Souslin space. If V is a closed and absolutely convex neighborhood of zeru in F, T- 1 v
=
prE [G(T)n(ExF)]
'
where prE is the projection on E. The space E x F is a Souslin space; G(T)
n
(ExV) is a sq-Borel subset, hence another Sous-
lin space, and its projection is again a Souslin space. Moreover T- 1 v is absorbent in E, hence it is non meagre. By cor.VII.3.8, it is thus a neighborhood of zero, hence T is continuous. REMARKS. We may assume that ~(T) is non meagre in E (which implies that E is a Baire space) and conclude that ~(T)
=
E.
The result can also be formulated for linear relations and thus includes the analogous open
ma~ping
theorem. These improv-
ments are straightforward and will not be developped here.
137
The theorem of Schwartz is restrictive in the sense that all range spaces must be separable (because they are Souslin spaces). It is clear that the completing webs, by relaxing the definition of a seave, remove that restriction. We quote now another proof, due to Christensen [13], replacing the topological developments of §3 by a group theoretical argument. Let us first develop some further properties of Souslin spaces. DEFINITION VII.4o2o Let X be a Souslin space. A strict seave of X is a seave e (v~I) such that, for each strand e v
vk
( k ~lN),
00
n
e
k=1
vk
~
p •
If x is the limit of a strand e oo
n
it is clear that
k= 1
!X}
e vk
vk
(k ~lN) of a strict seave,
o
PROPOSITION VII.4.3. Every Souslin space admits a strict seave. Let e (v~I) be a seave of the Souslin space X. Define S v
the collection of all strands vk(k ~lN) of I such that v 1 and
as
v
>
v
00
e
u
v
n
vk(k ~lN) ~ Sv k=1
e vk
C e C e • Indeed, if x ~ e , there exists a strand v v v v (k ~lN) ~ S such that x ~ e for each k ~ m. The second vk vk inclusion is obvious.
We have e e
The subsets ; Indeed, let Moreover, if
X
v
~
(v ~I) define a strict seave of X. ~
I
E- e ' ~
be given. If v
>
!.It
there is a strand e
S vk
v (k
C S
1.1 ~JN)
thus e ~
s
v
C e
such
~
00
that X
~
n k=1
X
~
1 38
e
vk
Hence e
• The strand e
J.l
ev•
vk
(k
> 1)
belongs to S
v1
,
thus
~
•
Take now a strand
evk (kE-lN).
converges to the same limit as e
vk for each k , hence the result.
-
C e
X f
C e C e , it vk vk vk (kE-lN). If x is that limit,
Since e
0
vk 0
PROPOSITION VII.4.4.
Let X be a Souslin space.
a strict seave of X, each e --
1L
e (vE-I) ~ v
is a Souslin space.
I.L
-
Indeed, e (v>~.L) is a seave of e • Indeed, it is a web of e • I.L
v
x f e
vk
vk for each k, hence x f
PROPOSITION VII.4.5.
I.L
( k E-lN) converges to x, we have
More over, if the strand e
e.
be a regular Hausdorff topological
~X
space and e a Souslin subspace of X. Then e is measurable with respect to all Borel measures on X. We prove prop.VII.4.5 only for a bounded and positive Borel measure I.L on X. Let e (vE-lN) be a seave of e. We have seen in prop.VII.1.2 v
that the map
q> : nk(k flN) -
lim ec
is continuous of lNJN lim ec
n1, ••• ,nk
n 1 ' • • • 'nk
)
onto e. Moreover,
""
n
) =
k=1
for all nk ( k flN) E- lN lN • Let Ik(k flN) be finite subsets of lN. The subset K
""n k=1
is compact in
k
n
-
u
i=1 n.E-I. l.
x.
e ( n 1 ' • • • 'nk)
l.
Indeed,
00
K•
n ri
i=1
is compact in IN:N • It is thus enough to prove that
(K') C K.
Let us check the converse. Given 1 39
x
~
K,
there exists, for each k, a finite sequence
(nk, 1 , ••• ,nk,k) such that
x ~
;:;< nk ,
1 , ••• , n k , k
)'
~(k)
We can choose an element IIi
~~k) ~ l.
Since
I.
l.
~
k
-.( k
convergent subsequence n all i
~
JN 0
of K 1 such that
o
for all k
For each k and for k
~
vk
I )
~
i,
admits as
which converges to some n
large enough, we
1
~(k)
the sequence
-.( k have n 1
I )
K1 •
~
n.
l.
for
k.
Thus x~ec
and
nk
I f
1
f o o o f
nk
I f
k
I
)(e(
nk
I f
1
f • o o f
nk
I f
k
)
00
hence the result. Let now and let
£
~
>
be a
bounded and positive Borel measure on X
0 be given. Denote by
~
the set of all Borel sub-
sets of X and define A.(A)
=
inf
f~(B)
: B ~~and B ) A}
for every subset A of X.
By the theorem of Levi,
B ~ ~ such that B ) A and ~(B)
=
there exists
A.(A).
If Am T A , A.(A) ~ sup A.(Am) m Indeed,
•
choose B ~ ~ such that B ) A and ~(B ) m m m m
We may assume that Bm T.
1 40
Indeed, since
A.(A ) • m
A(Am) ~ ~(B mnB m+ 1 ) ~~(B) m
,
it follows that ~(B \ B 1 ) = o, hence we can replace B 1 by m m+ m+ Bm U Bm+ 1 and thus, by induction, obtain an increasing sequenc e B 1 ( m HN ) • Now , m 00
sup A(Am) = sup ~(Bm)
m
~(
m
U Bm) ~ A(A) m=1
,
00
because A C
U
Bm•
m=1 Since 00
e
=
U
n 1 =1 there exists thus N1 such that N1 e ) + e/2 , A(e) ~ A( U n1 n 1 =1 and, by induction, a sequence Nk(k E-lN) such that Ni A( n u e(n1, ••• ,nk) i=1 n.=1 l. k+ 1 Ni k
~
A( n
u
e(
i=1 n.=1 l.
n,, ••• ,nk+1
)) +
£
I 2 k+1 ,
lfk f lN.
Thus, by summing up, Ni
k
A(e) ~
A(i~1 n~=1
e(n,, ••• ,nk))
+e,lfkE-lN.
l.
Therefrom, if e 0 E- ~ is such that e 0 have
~ ( e 0) ~ ~ (
k
Ni
n
u
i=1 n.=1
e(n1, ••• ,nk ) )
e and ~(e 0 )
)
+
£.
A(e), we
ltk E- lN '
l.
hence, by the theorem of Levi,
~ ~(
00
k
Ni
n
n
u
k=1 i=1 n.=1 l.
e( n , •.•• , nk )) + 1
£
•
141
Since Ni
k
co
n
n
e( n 1 , ••• ,nk ) c
u
k= 1 i= 1 n.=1
c
e
l.
~·
it follows that e is measurable with respect to PROPOSITION VII.4.6.
~
X be a compact group and
~
the norma-
lized Haar measure of X. If e C X is measurable with respect
!s
~ and such
that ~(e)
>
O, ~ e
-1
.e is a neighborhood of
the unit element of X. Consider the convolution product oe
*
b _1• e
If
£
is the unit
element of X,
Since be of
£
*
o _ 1 is continuous, there exists a neighborhood w e such that oe * o _ 1 > 0 on w. Thus, for each e
y f w, oe(x) o _ 1 (yx- 1 ) is not vanishing identically e exists x f
e such that yx
-1
f
e
-1
, hence y f
e
-1
there
.e and w C e
-1
.e.
Another proof of prop.VII.4o1o By the reductions already developped, we may assume that E is a Banach space. The map T will be continuous if, for every sequence x (mflN) f E such that
l'x
m
~ 2-m,
the sequence Tx m (m flN) is bounded in F. Consider the compact group X= {0,1}lN and the continuous 11
m~~
map
co X
of X into E.
Let
~
n
be the normalized Haar measure of X. If V
is a closed neighborhood of zero in F, (To~)- 1 lin subspace of X for each n fm.
(nV) is a Sous-
Indeed, X is a separable com-
plete metric space, hence a Souslin space. The graph G(T) of T is a sq-Borel subset of E x F. Its inverse image by the continuous map (~,I) is thus a Borel subset of X x F. (Observe that
142
the class of subsets e of E x F such that (~,I)- 1 e is a Borel set contains the sequentially closed subsets of E x F and is stable under countable intersections, countable unions and complements). But this inverse image is precisely the graph of To~.
Thus G(To~) is a Souslin space and
(To~)- 1 (nV)
prX[G(To~) n (XxnV)]
is again a Souslin space.
(To~)- 1 (nV) is thus measurable with
By prop.VII.4.5, en
00
respect
U en' hence there exists n ~m n=1 _1 such that ~(en) > o. By prop.VII.4.6, en .en is then a neigh1 borhood of zero in X. Thus b mk(k ~m) ~ en .e n for m large to~·
Moreover X
enough. If b k(k ~m)
m
=
f1
m(m E-lN) - fl'm(m ~lN) , m for m
f1 1 00
00
xk =
l:
m=1
flm
X
m
l:
m=1
fl'm
X
m
-/=
k. Thus
E- 2n T- 1 v
for k large enough, hence Txk ( k ~lN) is bounded in F. 5. THE BOREL GRAPH THEOREM FOR TOPOLOGICAL GROUPS We proceed now to the extension of the results of §4 to topological groups. The vector space structure was hardly used in Martineau's proof of prop.VII.4.1; more precisely, it was just needed in order that neighborhoods of zero be absorbent and through prop.VII.3.6. A slight modification of these arguments completely avoids the vector space structure. PROPOSITION VII.5.1. 1£1 E ~ F be topological groups and T an algebraic homomorphism of E into F. Assume that E is a Baire space,
~
E
~
F are Souslin spaces and that the graph of T
is a sg-Borel subset of E x F.
~
T is continuous.
The result is still true if T is only defined on a non meagre subgroup ~(T) ~ E and yields that ~(T) is open in E. 143
Let e (vE-I) be a strict seave of F. For each v E- I, e v
v
itself is a Souslin space (prop.VII.4.4)• From ~(T)
u
T
-1
e
vE-l
v
and T
-1
e
T
v
it follows that there exists a strand e T
-1
-1
vk
e
1.1
(k E-lN) such that
e
is non meagre for each k E-m. vk Moreover, T- 1 e
=
vk
prE [G(T)n(Exe
vk
)]
is a Souslin space. By prop.VII-3.7, 0 ( T -1 e
)
vk
\ T -1 e
vk
is thus
) =/=¢, otherwise vk itself would be meagre. Thus, by prop.VII.)o5o
meagre. However, by prop.VII.).), O(T- 1 e e
vk
O(T- 1 e hence T - 1 e
).o(T- 1 e
vk
vk
)- 1
vk
.(T- 1 e
vk
)- 1 J.·s a neJ.g · hb or h oo d o f
th e unJ."t e 1 emen t
e: of E.
Since T is an algebraic homomorphism, T -1 e
vk
.(T -1 e
vk
) -1 = T -1( e
vk
.e -1) • vk
.e- 1 (kE-lN) (k E-lN) is convergent in F, hence e vk vk vk is convergent to the unit element e:' of F.
The strand e
Thus, for every neighborhood w of e:' in F, T -1 w ) T -1( e
vk
)-1 .e -1) ) O(T -1 e ). O(T -1 e vk vk vk
for k large enough, hence it is a neighborhood of e:. Therefrom T is continuous and ~(T) is a neighborhood of e:. Since it is a subgroup, it is then open in E.
144
COROLLARY VII.5.2.(28]. If a topological group X is a Baire space and a Souslin space, it is a separable complete metric space. Since X is a Souslin space, it is separable. Let e (v~I) be a strict seave of X. By the proof of prop. v
VII.5.1 applied to the identical map of X into itself, there exists a strand e (k ~lN) such that e .e- 1 (k ~JN) is a base of vk vk vk neighborhoods of E in X. Thus X is metrizable. Let now
X be
the completion of X (as a topological group).
X.
The space X is non meagre in the identical map I of domain ~(T)
Xx
X. Thus X
X into
Apply again prop.VII.5.1
to
X : it is an homomorphism, its
X is non meagre in X and its graph is closed in ~(T) is open in X. Since X is a subgroup, it
contains the connected component e
0
of
dense. So, for each connected component e open, X contains some x
~
e0
,
X. It is moreover of X, since e is
in
E
hence it contains x.e
0
e.
Comment. We may thus reduce the apparent generality of prop. VII.5.1 where the domain space E is always a separable complete metric group. we might have proved
It is true in particular for l.c.s.
prop.VII.4.1 when the domain space is an inductive limit of Baire Souslin spaces. But these are
~-spaces,
hence we are led
back to the ultrabornological case. 6. FURTHER PERMANENCE PROPERTIES OF SOUSLIN SPACES We have proved in prop.VII.5.1
(and already used in its corol-
lary) a localization property similar to prop.IV.6.1. We quote it only for l.c.s., where we intend to use it to prove new permanence properties. PROPOSITION VII.6.1.
~
E be a separable
l.c.s. and ev(v ~I) a strict seave of F.
~-space,
F a Souslin
l l T is a continuous
linear map of E into F, there exists a strand e (k ~ I) ~ -vk · T- 1 e 1"s a ne1g · h__ 1s non meagre an d that T- 1 e th a_ t T- 1 e vk vk vk 145
borhood of zero for each k e
vk
( k ~lN) with every e
~
m.
We may start the strand
such that T - 1 e
is non meagre. v1 v1 This result provides a way of exhibiting seaves in spaces
of linear maps. ~
PROPOSITION VII.6.2.
E be a separable V-space or a coun-
table inductive limit of such spaces and F a sequentially complete Souslin l.c.s. or a countable inductive limit of such spaces. ~ Lpc(E,F) is a Souslin space. Assume first that E is a separable V-space and F a sequentially complete Souslin space. Denote by Uk(k ~m) a base of neighborhoods of zero in E, by {xi : i ~lN} a countable dense subset of E and by e (v~I) a strict seave of F. v
Define the subset&(.
l.t!ltV
T ~ L(E,F) such that T- 1 e
ll
) of L(E,F) as the set of all
is non meagre, TU. C e - e and l. ll ll
Tx 1 ~e. Given&(.l.tjltV )'define&(.l.t!ltV ) t (•Jtil 1 tV 1 t~r)(ll'>ll,v'>v) V ~
as the set of all T TU. C e J
e
-
Tx 1
1 ,
&(.
l.t!ltV
~
e
1
) such that T ~
and Tx 2
-1
e .. , is non meagre, ,.
er •
ll 1 ll v ~ Continuing this procedure and relabeling at each stage by
an index in
m,
we clearly define a web in L(E,F).
A strand
and strands e that, i f T TU. l.k
~
c
e
Ilk 'ef'k Ilk
of this web determines a sequence ik(k ~m)
(k ~m) and e
vi,k
(k ~m) of the seave of F such
'
-
e
Ilk
and
Txi
~
e
vi,k
,
Vi
~
k •
sequence Tk is thus equicontinuous. MoreoIf Tk ~ 'cf'k ' the ver, for each i ~ lN, Tk xi converges to lim e • By the k.......,., vi,k Banach-Steinhaus theorem, the sequence Tk(k ~m) is thus convergent in Lpc (E,F) and its limit T depends only on 'ctk(k ~m), thus
~
146
L(E,F) maps E into some Fi, hence
00
L(E,F) ..
U L(E,F.). ~ i=1
Moreover, the topology of each Lpe (E,F.) is finer than the to~ pology induced by Lpc (E,F), hence the result, by prop.VII.2.1 and VII.2.3. If now E is the inductive limit of a sequence of separable u-spaces E. (i E-lN), L(E,F) can be identified to a closed sub~
space L of
n
i= 1
L
pc
(E.,F) and the canonical map of L equipped ~
with the topology induced by
n i=1
Lpc (E.,F) into Lpc (E,F) is ~
sequentially continuous (cf.prop.VII.2.11). Hence the result by prop.VII.2.B. PROPOSITION VII.6.}.
~
E be a metrizable Schwartz space and
F a sequentially complete Souslin l.c.s. or a countable inductive limit of such spaces. ~ Lb(E{,,F) is a Souslin space. Denote by
E
the completion of E. It is still a Schwartz spa-
ce, thus, by cor.III.}.10, bounded subset of
E
(E){,
is ultrabornological. Every
is relatively compact hence, by prop.
III.1.4, it is contained in the closed absolutely convex hull
(E){,
of a convergent sequence of E. Therefore to
is isomorphic
E{,o If Um(m E-lN) is a base of neighborhoods of zero in E, E{, is
the inductive limit of the Banach spaces E 1
•
We can choose
u~
~
the Ui's in such a way that Ui is compact in E' • Then E{, is Ui+1 still the
induc~ive
limit of the separable Banach spaces obtai-
ned by equipping the closure of E'
U o.
~
E'
in E 1
uo
with the norm of
i+1
Ui+1 Observing finally that every bounded subset of E{, is precompact, Lb(E{,,F)
=
Lp 0 (E{,,F) and we can conclude by prop.VII.6.2.
147
~
PROPOSITION VII.6.4.
E be the strict inductive limit of a
sequence of metrizable Schwartz l.c.s. Ei(i ~m) ~ F be a sequentially complete Souslin l.c.s. of class Q. ~ Lb(Eb,F) is a Souslin space. See def.V.1.7 for the definition of Q. We have proved in prop.V.1.6 that 00
=
L(E 1 ,F) C
U ~.L(E! ,F) i= 1 1 1,c
where ' i is defined by
h.T)x' 1
T(x'IE.) • 1
Since the spaces E.(i ~m) are metrizable Schwartz spaces, 1
E1 E 1 and E! = E!1,b • Moreover, since every bounded subset C b 1,c of E is contained and bounded in some Ei, each •i is continuous of Lb(E!
1,c
,F) into Lb(E',F). Hence the conclusion, by C
prop.VII.6.e. ~
PROPOSITION VII.6.5.
E be a Schwartz ...
plete Souslin l.c.s. Then E
~
-
£
~-space
and F a com-
F is a Souslin space.
Indeed, we have seen in prop.V.2.1
that E
~
£
F is a closed
subspace of Lr(E' ,F). Since E is a Schwartz ~-space, it is ca moreover clear that Lr(E~a,F) = Lb(Eb,F), hence the result by prop.VII.6.:5. PROPOSITION VII.6.6.
~
sequence of Schwartz
~-spaces
of class Q. Then E
~
-
£
E be the strict inductive limit of a and F a complete Souslin l.c.s.
F is a Souslin space.
Same proof, using prop.VII.6.4. PROPOSITION VII.6.7.
~
E be a separable 9'-space or a counta-
ble inductive limit of such spaces and F a complete Souslin l.c.s. ~ E~a ® F is a Souslin space.
...
We have seen in prop.V.2.4 that E 1 ® F is a closed subspaca £ ce of Lca (E,F), which is a Souslin space by prop.VII.6.2. 148
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Index
Absorbing disk Algebraic closure
34 44
Almost open
10
Associated -bornological (ultrabornological) topology
................
Complete fast
Completing -disk ••o• -sequence
... 0
sequence o•o•
strictly
-&-topology
26
-web
-relation
13
Component
-sequence
49 49
Continuous relation
-strictly
sequence
Baire space
1
...
-theorem
2
Banach-Dieudonne theorem Banach disk Barrel
.....
.............
3 3
Barrelled space
......... Borel set ....
Borel sq
6 29
34
Bornological
35
Bounded fast- •••••
...
locally-
...
fll.-
....... Closure algebraic Compact fast
............ ................
37 37 73 37 73 44 30
•
Convergent Mackeyfastc (E) 0
C(K,E) Disk
0
.....
•
15 13 29
•
0
29
97 97
•
0
•
..... .....
... ....... ... .... ...... .... ................
Evaluable &(g,E) 0 0
EJ'F B'u: E'me ' E'fc
49 49
••. 29,34
•
Domain
.....
.............
...... ..... .......
130
Bornivorous
0
29
48
•
35
0
92
Factor subspace Fast -bounded -compact
......... ...
....
13
14 19 21
133
44 97 19 21
37 15
37 29
157
Fast -complete -convergent
............ ..........
-sequentially closed Homomorphism
...........
Inductive class Infra-Schwartz Inverse relation
........ ......... .....
Isomorphism weak
•••
••••••••••••••.••
.. ........ ........ ................ ................ ... .................. ................
Krein-Smulian theorem
29 19
26 43 12 107 107 8
12
Locally bounded
31 124
L(E,F), Ls(E,F) ••••••••• l(E)
76 96
100 (E)
97 93 96 96
L00 (e,E)
2
Loo (em ( m E-lN 'E) • • • • • • • • • • L:(e,E), L00 (em(mE-lN)'E) Mackey -convergent
••••••••••
Norming disk
Open almostrelatively-
relatively weakly- ••••
1
29
Seave Strict
................... ..................
Simple subspace Souslin space Strand
•••••••••
..................
Strict -web -seave
.................. ................ ...........
Strictly -associated
-completing -webbed
............
...............
Strongly inaccessible Transpose
•••
............... ........
Ultrabornivorous
84
126 138
V-selection Web
15
49 49 138
49 49 49 40
1 34
•••••• !5,35
............
.................... ...............
103
................
25
••••••••••• 126
Well located
103
24
v- ..••................. 11e
103
133
73
Selection
completing- ••••••••••• strictWebbed strictly-
...................
Q(class) 158
29
10,13,103 10
weakly- •••••••••••••••
O(e)
fll.l/ , ••••••••••••••••••
Ultrabornological
................. ........... ............... ...........
Meagre
Rare Reflexive web
10
Linear relation Located well-
................... ................... . .......... . ......................
Range
92
............ ...........
118
48
49 49 49 124
