Table Of ContentZoran Kadelburg Stojan Radenovi´c
SUBSPACES AND QUOTIENTS OF
TOPOLOGICAL AND ORDERED
VECTOR SPACES
Novi Sad, 1997.
CONTENTS
INTRODUCTION..................................................... 1
I: TOPOLOGICAL VECTOR SPACES............................... 3
1.1. Some properties of subsets of vector spaces....................... 3
1.2. Topological vector spaces......................................... 6
1.3. Locally convex spaces............................................ 12
1.4. Inductive and projective topologies............................... 15
1.5. Topologies of uniform convergence. The Banach-Steinhaus theorem 21
1.6. Duality theory................................................... 28
II: SUBSPACES AND QUOTIENTS OF TOPOLOGICAL VECTOR
SPACES......................................................... 39
2.1. Subspaces of lcs’s belonging to the basic classes................... 39
2.2. Subspaces of lcs’s from some other classes........................ 47
2.3. Subspaces of topological vector spaces............................ 56
2.4. Three-space-problem for topological vector spaces................. 60
2.5. Three-space-problem in Fr´echet spaces............................ 65
III: ORDERED TOPOLOGICAL VECTOR SPACES ................ 72
3.1. Basics of the theory of Riesz spaces............................... 72
3.2. Topological vector Riesz spaces................................... 79
3.3. The basic classes of locally convex Riesz spaces................... 82
3.4. l-ideals of topological vector Riesz spaces......................... 90
3.5. 3SP in the class of locally convex Riesz spaces.................... 92
IV: SUPPLEMENT: ENLARGEMENTS OF LOCALLY CONVEX
TOPOLOGIES.................................................. 100
4.1. Finite-dimensional enlargements.................................. 100
4.2. Countable enlargements of barrelled topologies................... 103
4.3. Satisfactory subspaces of barrelled spaces......................... 106
4.4. Open problems................................................... 112
REFERENCES........................................................ 113
SUBJECT INDEX.................................................... 120
INTRODUCTION
First results concerning not necessarily normed topological vector spaces date
back to middle thirties of this century. But it was not until late forties and early
fifties that the theory of such spaces was formed as a special branch in functional
analysis. This was the time when it became clear that a lot of applications,
in particular the theory of distributions, need more abstract approach than the
classical Banach one.
Withintheframeworkofthenewcategoriesoftopologicalvector,inparticular
locally convex, spaces and, a bit later, of ordered topological vector spaces, a lot
of special classes of spaces were introduced. The reason was mainly that the three
fundamental principles of linear analysis, in particular the uniform boundedness
principle and the closed graph and open mapping theorems, needed special types
of spaces to serve as domains and/or codomains for respective mappings. Such
classes of spaces, e.g. barrelled, quasibarrelled, bornological and many others,
provedthemselvestobeimportantinconnectionwithotherproblems, too, suchas
problems in duality theory, reflexivity, tensor products and so on.
One of the important questions concerning these classes of spaces was their
stabilityin connectionwith subspaces andquotients. While most ofthem (but not
all) are stable with respect to quotients by closed subspaces, few are stable with
respect to arbitrary, even closed subspaces. So it was natural to search for various
sufficientconditionswhichwouldimplystabilitiesofsuchkind. Firstresultsofthat
type were obtained in early fifties, but investigations have continued and there are
a lot of interesting recent results. Some questions even in pure Banach setting are
still open, e.g. “is there a Banach space of infinite dimension such that all of its
dense subspaces are barrelled” or “does every infinite-dimensional Banach space
have a separable quotient”?
This book is divided into four chapters. The first one contains a very brief
introduction to the theory of topological vector spaces. The intention was to give
only the fundamental ideas of the theory, particularly those which will be used in
the rest of the book. Some important theorems are stated without proofs, and
the reader is referred to the classical books covering these problems. But we hope
that this will be enough to get prepared for reading the remaining three chapters.
Especially, at least formally speaking, the prerequisites for the whole book are
just the basics of linear algebra and general topology, although, of course, the
acquaintance with classical functional analysis is desirable.
2 Introduction
Chapter II treats thoroughly the problems of subspaces and quotients for
various classes of topological vector, in particular locally convex, spaces. Also,
a kind of a converse question, the so-called three-space-problem, is investigated.
Some of the results of the authors are included in this and the following chapter.
In many applications of functional analysis, besides linear and topological
structures, also the structure of order is often important. So a special theory
of ordered vector spaces, and in particular ordered topological vector spaces, has
beendeveloped. Ourthirdchaptertreatsthebasicsofthistheoryandinparticular
thoseconceptswhichareclosetotheonesdescribedinthefirstpartofthebook. It
isshownthatthericherstructureenablesmorepositiveresultsthaninthecategory
of vector spaces without order.
There are a lot of problems and streams of investigation which are actual
nowadays in the theory of topological vector spaces. We chose some of them and
included as a supplement in the chapter IV. We hope that this will help to get
an idea in what directions the theory may be developed and, maybe, some of the
readers will try to do something on their own in this field. A brief list of open
questions which is given at the end might help.
Althoughthelistofreferencesisratherlong,itisveryfarfrombeingcomplete,
even when the special topics of this book are concerned. We mentioned just those
titles which we referred to directly and, with a few exceptions, we didn’t mention
articles which are already included in existing monographs.
We are very grateful to the Institute of Mathematics of the University of Novi
Sad for publishing this monograph.
Beograd, October 1996 Authors
I. TOPOLOGICAL VECTOR SPACES
1.1. SOME PROPERTIES OF SUBSETS OF VECTOR SPACES
All vector space in this book are supposed to be defined over the fixed field
K, which is either the field R of real or the field C of complex numbers. Only in
the third chapter just real scalars are used.
Inthisparagraphweshalldescribesomepropertiesofsubsetsofvectorspaces
which are important in the theory of topological and ordered vector spaces.
Let E be a vector space (over the field K) and let A be its subset. We say
that A is:
1◦ circled if λA⊂A for each λ∈K for which |λ|(cid:54)1;
2◦ convex if λx+(1−λ)y ∈A for each x,y ∈A and each λ∈K, 0(cid:54)λ(cid:54)1;
3◦ absolutely convex if λx+µy ∈ A for each x,y ∈ A and each λ,µ ∈ K for
which |λ|+|µ|(cid:54)1.
It is clear that the intersection of an arbitrary family of (absolutely) convex
subsets is again an (absolutely) convex set. The (absolutely) convex cover of the
set A ⊂ E is the intersection of all (absolutely) convex sets containig A; we shall
denote it by coA (resp. ΓA). If (A ), i ∈ I is a family of subsets of E, we shall
i (cid:83) (cid:83)
write co A (resp. Γ A ) instead of co( A ) (resp. Γ( A )).
i i i i i i i i
(cid:80)n
Proposition 1.1. coA is the set of all elements of the form λ x for
i i
1
(cid:80)n (cid:80)n
λ (cid:62) 0, λ = 1, x ∈ A. ΓA is the set of all elements of the form λ x for
i i i i i
1 1
(cid:80)n
λ ∈K, |λ |(cid:54)1, x ∈A.
i i i
1
Proof. We shall prove the assertion for the convex cover; for the absolutely
convex cover the proof is similar. It can be checked directly that the elements of
the given form create a set which is convex and contains A. Let us show that,
conversely, these elements belong to each convex set which contains A.
n(cid:80)−1
Suppose that the last assertion is proved for each sum of the form λ x .
i i
1
(cid:80)n n(cid:80)−1
Let µ x be a combination of the given form and let µ =µ>0 (for µ=0
i i i
1 1
4 I.Topologicalvectorspaces
n(cid:80)−1µ
the assertion is trivial). By the inductive hypothesis, y = i x ∈coA, and so
µ i
1
(cid:80)n
µy+(1−µ)x = µ x ∈coA since 0<µ(cid:54)1.
n i i
1
By the circled cover of a set A ⊂ E we understand the intersection of all
circled subsets of E which contain A.
Proposition 1.2. The absolutely convex cover of a subset is the convex cover
of its circled cover. A set is absolutely convex if and only if it is convex and circled.
The circled cover of the convex cover of a set need not be convex.
Proof. Since, obviously, each absolutely convex set is circled, it is enough to
prove that the convex cover of a circled set A is absolutely convex. According to
Proposition1.1,wehave coA⊂ΓA. Conversely,let λ (cid:54)=0 from K begiven,such
i
(cid:80)n (cid:80)n (cid:80)n
that |λ |(cid:54)1 andlet x ∈A. Denote µ =(λ /|λ |) |λ | and ν =|λ |/ |λ |.
i i i i i i i i i
1 1 1
(cid:80)n (cid:80)n
Then each x = |λ |x ∈ ΓA can be written in the form x = ν µ x , where
i i i i i
1 1
(cid:80)n
ν =1, ν (cid:62)0 and |µ |(cid:54)1, µ x ∈A, so that x∈coA. Thus ΓA⊂coA.
i i i i i
1
Let A = {(0,0),(1,0),(1,1)}, treated as a subset either of R2 or C2. It is
easy to check that the circled cover of the set coA is not convex.
Proposition 1.3. 1◦ If A , ... , A are (absolutely) convex sets and λ ,
1 n 1
... , λ ∈K, then λ A +···+λ A is an (absolutely) convex set.
n 1 1 n n
2◦ If E and F are vector spaces over K and L: E →F is a linear mapping,
then the image (resp. inverse image) of each (absolutely) convex subset of E
(resp. F) is an (absolutely) convex subset of F (resp. E).
Proof. The assertion 1◦ can be checked directly. Let A be a convex subset
of E. Since L(λx+(1−λ)y)=λLx+(1−λ)Ly, the set L(A) is convex in F if
A is a convex subset of E. Similarly, if B is convex in L(E) and if Lx, Ly ∈B,
then L(λx+(1−λ)y) ∈ B and so L−1(B) is a convex set in E. For absolutely
convex subsets the proof is similar.
Let A and B be subsets of a vector space E. We say that A absorbs B if
there exists α>0, such that B ⊂λA for each λ∈K such that |λ|(cid:62)α. The set
A is absorbing in E if it absorbs all the points, more precisely all the singletons
from E, i.e. if for each x∈E there is α>0 so that x∈λA for each λ∈K such
that |λ|(cid:62)α. It can be immediately checked that the following is valid.
Proposition 1.4. Let A and B be circled sets. Then A absorbs B if and
only if there exists λ>0 such that λB ⊂A. In particular, the set A is absorbing
if and only if for each x∈E there exists λ>0 such that λx∈A.
We call a sequence U = (U ) of subsets of the vector space E a string if all
n
the subsets U are circled and absorbing and U +U ⊂ U holds for each
n n+1 n+1 n
1.1. Somepropertiesofsubsetsofvectorspaces 5
(cid:84)
n ∈ N. The set U is the n-th knot of this string and N(U) = U is its
n n∈N n
kernel — it is obviously a vector subspace of the space E.
To each absorbing absolutely convex set U one can naturally correspond a
string U = (U ) by U = (1/2n−1)U, n ∈ N. However, in general, knots of a
n n
string need not be convex.
Example. Let E be a vector space and {e } its algebraic (Hamel) basis.
α α∈A
For a positive real number p let us define a string Up =(Up) by
n
(cid:80) (cid:80)
Up ={x|x= λ e , |λ |p (cid:54)1/2n−1}, for 0<p(cid:54)1,
n α α α
α∈A α∈A
(cid:181) (cid:182)
(cid:80) (cid:80) 1/p
Up ={x|x= λ e , |λ |p (cid:54)1/2n−1}, for p(cid:62)1,
n α α α
α∈A α∈A
where in all given sums we assume that λ (cid:54)= 0 only for finitely many indices α.
α
For each p we have N(Up)={0}. For p(cid:62)1 the knots Up are obviously convex,
n
but for 0<p<1 it need not be true. More precisely, the following is valid.
Proposition 1.5. Let 0 < p < 1. Then the following conditions are
equivalent:
(i) E has an uncountable algebraic dimension;
(ii) no knot Up of the string Up contains an absorbing absolutely convex set.
n
Proof. (i) =⇒ (ii). Let (e ) beabasisof E. Itisenoughtoprove(ii)for
α α∈A
n=1. If there exists an absolutely convex and absorbing set U ⊂Up, then there
1
exist ε >0 such that ε e ∈U for all α∈A. As A is uncountable, we can find
α α α
ε > 0 such that the set A := {α ∈ A | ε (cid:62) ε} is infinite. If α ,...,α ∈ A ,
ε α 1 n ε
ε (cid:80)n ε
we have ε e ∈ U and e ∈ U. On the other hand, U ⊂ Up and so
ε αi αi n αi 1
(cid:179) α(cid:180)i i=1
(cid:80)n ε p
1(cid:62) =n1−pεp. Since n is arbitrary, this is a contradiction.
n
i=1
(ii) =⇒ (i). Let us show that in the countable case each string U = (U )
n
contains an absolutely convex string. Let {e } be an algebraic basis of the
n n∈N
space E. If we choose ε(1) > 0 with ε(1)e ∈ U , ε(n) with 0 < ε(n) (cid:54) 1ε(n−1)
i i i 1+i i i 2 i
(cid:80)∞
and ε(n)e ∈ U for n > 1, then the sets V := ε(n)[e ] form an absolutely
i i n+i n i i 1
i=1
convex string V for which V ⊂U (i.e. V ⊂U for each n). Namely,
n n
(cid:80)∞ (cid:80)∞
V = ε(n)[e ] ⊂ U ⊂U .
n i i 1 n+i n
i=1 i=1
(cid:80)
Here we used the following notation: if M ⊂ E, α ∈ A, then M is the set
α α
(cid:181) (cid:182) α∈A
(cid:83) (cid:80)
M , where e runs through all finite subsets of the index set A. If
e⊂A α
α∈e
x∈E and λ>0, [x] :={µx||µ|(cid:54)λ}.
λ
6 I.Topologicalvectorspaces
We shall assume that the concepts such as subspace, quotient, (algebraic)
dimension and codimension are known to the reader. Particularly, we shall
call subspaces of a vector space E which have the codimension 1 hyperplanes
(sometimes we shall use this name also for linear manifolds which are obtained by
their translation for a fixed vector).
If E is a vector space over K, then the set of all linear mappings f: E →K,
with linear operations canonically introduced, is called the algebraic dual space of
the space E and is denoted by E∗. An obvious connection between the algebraic
dual and hyperplanes is given by
Proposition 1.6. To each element f, not identically equal to zero, of the
algebraic dual E∗ of a vector space E, a hyperplane H = f−1(0) of this space is
corresponded. Conversely, to each hyperplane H of a space E one can correspond
an element f ∈E∗ and α∈K such that H =f−1(α).
1.2. TOPOLOGICAL VECTOR SPACES
Definition 1. Let E beavectorspaceoverthefield K. Topology t on E is
calledvector (linear)ifitiscompatiblewithlinearoperations,i.e.if (x,y)(cid:55)→x+y
and (λ,x)(cid:55)→λx arecontinuousmappingson E×E and K×E,respectively. The
space E equipped with a Hausdorff linear topology t on it, i.e. the ordered couple
(E,t) is called a topological vector space (tvs) or linear topological space.
An immediate consequence of the definition is the following
Proposition 2.1. Let (E,t) be a tvs, x ∈E and α∈K. Then:
0
1◦ themapping x(cid:55)→x+x isatopologicalhomeomorphismofthespace (E,t)
0
onto itself;
2◦ if U is a base of neighbourhoods of the point 0∈E, then the sets x +U,
0
U ∈U form a base of neighbourhoods of the point x ;
0
3◦ a linear mapping f: E →F between tvs’s is continuous on E if and only
if it is continuous at the point 0.
Proposition 2.2. Each neighbourhood of the origin in a tvs (E,t) is an
absorbing set. The space (E,t) possesses a base of neighbourhoods of the origin
formed of circled subsets and to each circled neighbourhood of the origin U there
corresponds a string U =(U ), such that U =U and all U ’s are neighbourhoods
n 1 n
of the origin (such a string is called topological). The intersection of all the
neighbourhoods of the origin is the set {0}.
Proof. Ifaneighbourhood U oftheoriginin (E,t) werenotabsorbent,apoint
x ∈E would exist which does not belong to any nU, n∈N. Then 1x ∈/ U for
0 n 0
all n∈N would be valid, which is contrary to the condition 1x →0, which is a
n 0
direct consequence of the continuiuty of the multiplication by scalars.
1.2. Topologicalvectorspaces 7
The continuity of the function λx at the point (0,0) also implies that for
each neighbourhood of the origin U there exist δ >0 and a neighbourhood of the
(cid:83)
origin W such that λx∈U for all x∈W and all λ, |λ|(cid:54)δ. Thus, V = {λW |
|λ|(cid:54)δ} is a neighbourhood of the origin which is obviously circled and contained
in U.
The existence of a string (U ) with the described properties is a consequence
n
of the continuity of the map (x,y)(cid:55)→x+y. The last assertion of the proposition
follows because the topology t is Hausdorff.
Letusshownowthatinacertainsensetheconverseofthepreviousproposition
holds.
Theorem 2.3. Let F be a family of strings on a vector space E such that:
for each two strings U,V ∈ F there exists W ∈ F with W ⊂ U ∩V
(i.e. W ⊂U ∩V for each n∈N).
n n n
Then the knots of all the strings from F form a base of neighbourhoods of the
origin of a linear topology t on E (this topology will be denoted as t and we shall
F
say that it is generated by the family F). The topology t is Hausdorff (i.e. (E,t)
(cid:84) F
is a tvs) if and only if N(U)={0}.
U∈F
Proof. Let U(0) be the family created of all the knots of the strings from F.
If the family U(x ) for a point x ∈ E is defined as U(x ) = {x +U | U ∈
0 0 0 0
U(0)}, it is clear that all sets from the families U(x ), x ∈ E will form a base
0 0
of neighbourhoods of a certain topology on E. Let us prove that this topology is
linear.
The continuity of the mapping (x,y) (cid:55)→ x+y at the point (x ,y ) follows
0 0
from (x +V)+(y +V) ⊂ x +y +U, where V ∈ U(0) is chosen such that
0 0 0 0
V +V ⊂U ∈U(0).
Supposethataneighbourhood λ x +U ∈U(λ x ) isgivenandlet |λ |(cid:54)n∈
0 0 0 0 0
N. Wecanfind V ∈U(0) suchthat V +V +···+V ⊂U;thenalso nV+V+V ⊂
(cid:124) (cid:123)(cid:122) (cid:125)
n+2
U. Chooseaninteger m suchthat x ∈mV . Ifnow |λ −λ|<1/m and x∈x +V ,
0 0 0
then the relation λx∈λ x +V +V +nV ⊂λ x +U follows from
0 0 0 0
λx=λ x +(λ−λ )x +(λ−λ )(x−x )+λ (x−x )
0 0 0 0 0 0 0 0
since 1(mV)=V , (λ−λ )(x−x )∈ 1V ⊂V and λ (x−x )∈nV . In such a
m 0 0 m 0 0
way we have proved that the mapping λx is continuous at the point (λ ,x ).
0 0
The last assertion can be checked immediately.
Examples. 1◦ The simplest examples of tvs’s are normed, in particular
Banach spaces. In such a space, a base of neighbourhoods of the origin is formed
by the balls K ={x∈E |(cid:107)x(cid:107)<1/n}.
n
2◦ The family of all strings in a vector space E generates the topology on E
which is the finest linear topology on this space. It will be denoted as t .
b(E)
