Normed Linear Spaces

217 Pages·1973·4.106 MB·English

by  Maiilon M. Day

Mahlon M. Day
Normed Linear Spaces
Third Edition
Springer-Verlag Berlin Heidelberg New York
1973
Mahlon M. Day
University of Illinois, Urbana, Illinois, U.SA.
AMS Subject Classifications (1970): Primary 46Bxx· Secondary 46Axx
ISBN 3-540-06148-7 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-06148-7 Springer-Verlag New York Heidelberg Berlin
ISBN 3·540-02811-0 Second edition Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-02811-0 Second edition Springer-Verlag New York Heidelberg Berlin
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agreement with the publisher © by Springer-Verlag Berlin Heidelberg 1958, 1962, 1973 Library of Congress Catalog
Card Number 72-96940 Printed in Germany Monophoto typesetting and offset printing Zechnersche Buch
druckerei, Speyer Bookbinding Konrad TriItsch, Wurzburg
Foreword to the First Edition
This book contains a compressed introduction to the study of normed
linear spaces and to that part of the theory of linear topological spaces
without which the main discussion could not well proceed.
Definitions of many terms which are required in passing can be
found in the alphabetical index. Symbols which are used throughout
all, or a significant part, of this book are indexed on page 132*. Each
reference to the bibliography is made by means of the author's name,
supplemented when necessary by a number in square brackets. The
bibliography does not completely cover the available literature, even
the most recent; each PFlper in it is the subject of a specific reference
at some point in the text. 1
The writer takes this opportunity to express thanks to the University
of Illinois, the National Science Foundation, and the University of
Washington, each of which has contributed in some degree to the
cultural, financial, or physical support of the writer, and to Mr. R .. R.
Phelps, who eradicated many of the errors with which the manuscript
was infested.
Urbana, Illinois (USA), September 1957 Mahlon M. Day
Foreword to the Third Edition
The major changes in this edition are: Corrections and additions to II,
§ 5, and III, § 1, on hw* and ew* topologies; enlargement of III, § 2, on
weak compactness, V, § 1, on extreme and exposed points, V, § 2, on fixed
points, and VII, § 2, on rotundity and smoothness; added sections
III, § 5, on weak compactness and structure of normed spaces, and VII,
§ 4, on isomorphism to improve the norm, and Index of Citations.
Urbana, September 1972 M.M.D.
*
In the 3rd edition page 201.

Contents
Chapter I. Linear Spaces . . . . . . . . . . 1
§ 1. Linear Spaces and Linear Dependence . 1
§ 2. Linear Functions and Conjugate Spaces. 4
§ 3. The Hahn-Banach Extension Theorem 9
§ 4. Linear Topological Spaces. . . . 12
§ 5. Conjugate Spaces . . . . . . . 18
§ 6. Cones, Wedges, Order Relations. 22
Chapter II. Normed Linear Spaces. . . 27
§ 1. Elementary Definitions and Properties 27
§ 2. Examples of Normed Spaces; Constructions of New Spaces
from Old. . . . . . . . . . 32
§ 3. Category Proofs. . . . . . . . . . . . . . 38
§ 4. Geometry and Approximation. . . . . . . . 43
§ 5. Comparison of Topologies in a Normed Space. 45
Chapter III. Completeness, Compactness, and Reflexivity 53
§ 1. Completeness in a Linear Topological Space. 53
§ 2. Compactness . . . . . . . . . . . . 57
§ 3. Completely Continuous Linear Operators. . 65
§ 4. Reflexivity . . . . . . . . . . . . . . . 69
§ 5. Weak Compactness and Structure in Normed Spaces 72
Chapter IV. Unconditional Convergence and Bases. 78
§ 1. Series and Unconditional Convergence . . 78
§ 2. Tensor Products of Locally Convex Spaces 83
§ 3. Schauder Bases in Separable Spaces. 87
§ 4. Unconditional Bases . . . . . . . . . . 95
Chapter V. Compact Convex Sets and Continuous Function Spaces 101
§ 1. Extreme Points of Compact Convex Sets . . 101
§ 2. Fixed-point Theorems . . . . . . . . . . . . . . . . 106
VIII Contents
§ 3. Some Properties of Continuous Function Spaces . . . 113
§ 4. Characterizations of Continuous Function Spaces among
Banach Spaces. . . . 116
Chapter VI. Norm and Order 126
§ 1. Vector Lattices and Normed Lattices 126
§ 2. Linear Sublattices of Continuous Function Spaces 131
§ 3. Monotone Projections and Extensions . . 135
§ 4. Special Properties of (AL)-Spaces . . . . 137
Chapter VII. Metric Geometry in Normed Spaces 142
§ 1. Isometry and the Linear Structure . . . . 142
§ 2. Rotundity and Smoothness . . . . . . . 144
§ 3. Characterizations of Inner-Product Spaces 151
§ 4. Isomorphisms to Improve the Norm . . . 159
A. Rotundity, smoothness, and convex functions. 159
B. Superreflexive spaces 168
Chapter VIII. Reader's Guide 175
Bibliography . . 184
Index of Citations 197
Index of Symbols 201
Subject Index . . 205
Chapter I. Linear Spaces
§ 1. Linear Spaces and Linear Dependence
The axioms of a linear or vector space have been chosen to display
some of the algebraic properties common to many classes of functions
appearing frequently in analysis. Of these examples there is no doubt
that the most fundamental, and earliest, examples are furnished by the
n-dimensional Euclidean spaces and their vector algebras. Nearly as
important, and the basic examples for most of this book, are many
function spaces; for example, C[O, 1], the space of real-valued conti
nuous functions on the closed unit interval, B V [0, 1], the space of
functions of bounded variation on the same interval, I! [0, 1], the space
of those Lebesgue measurable functions on the same interval which
have summable plh powers, and A(D), the space of all complexvalued
functions analytic in a domain D of the complex plane.
Though all these examples have further noteworthy properties, all
share a common algebraic pattern which is axiomatized as follows:
(Banach, p. 26; Jacobson).
Definition 1. A linear space L over a field A is a set of elements
satisfying the following conditions:
(A) The set L is an Abelian group under an operation +; that is,
+ is defined from L x L into L such that, for every x, y, z in L,
(a) x+y=y+x, (commutativity)
(b) x+(y+z)=(x+y)+z, (associativity)
(c) there is a w dependent on x and y such that x+w=y.
(B) There is an operation defined from A x L into L, symbolized
by juxtaposition, such that, for A, f1 in A and x, y in L,
(d) A(X + y) = Ax + Ay , (distributivity)
(e) (A+f1)X=AX+f1X, (distributivity)
(1) A(f1 x) = (A f1) x ,
(g) 1 x = x (where 1 is the identity element of the field).
2 Chapter I. Linear Spaces
In this and the next section any field not of characteristic 2 will do;
in the rest of the book order and distance are important, so the real
field R is used throughout, with remarks about the complex case when
that field can be used instead.
(1) If L is a linear space, then (a) there is a unique element 0 in L
such that x+O=O+x=x and ,uO=Ox=O for all ,u in A and x in L;
(b) ,ux=O if and only if ,u=O or x=O; (c) for each x in L there is a
unique y in L such that x+y=y+x=O and (-1)x=y; (then for
z,xinLdefine z-x=z+(-1)x and -x=O-x).
(2) It can be shown by induction on the number of terms that the
commutative, associative and distributive laws hold for arbitrarily large
L
finite sets of elements; for example, Xi' which is defined to be
i::;n
Xl + (X2 + (. .. + xn)···), is independent of the order or grouping of terms
in the process of addition.
Definition 2. A non-empty subset £, is called a linear subspace of
L if £, is itself a linear space when the operations used in £, are those
induced by the operations in L. If X =!= y, the line through x and y is
the set {,ux+(1-,u)y: ,uEA}. A non-empty subset E of L isjlat if with
each pair x =!= y of its points E also contains the line through x and y.
(3) £, is a linear subspace of L if and only if for each x, y in £, and
each A in A, x + y and Ax are in £'.
Definition 3. If E, F ~ Land z E L, define
E+F={x+y:XEE and YEF}. -E={-X:XEE},
E+z={X+Z:XEE}, E-z=E+(-z), E-F=E+(-F).
(4) (a) E is flat if and only if for each x in E the set E - x is a linear
subspace of L. (b) The intersection of any family of linear [flat] subsets
of L is linear [either empty or flat J. (c) Hence each non-empty subset
E of L is contained in a smallest linear [flat] subset of L, called the
linear [jlat} hull of E.
Definition 4. If L is a linear space and Xl' ... , Xn are points of L, a
point x is a linear combination of these Xi if there exist Al ' ... , An in A such
L
that x = Ai Xi. A set of points E ~ L is called linearly independent
i::;n
if E is not-0 or {O} andl no point of E is a linear combination of any
finite subset of the other points of E. A vector basis (or Hamel basis)
in L is a maximal linearly independent set.
(5) (a) The set of all linear combinations of all finite subsets of a
set E in L is the linear hull of E. (b) E is linearly independent if and
1 0 is the empty set; {x} is the set containing the single element x.
§ 1. Linear Spaces and Linear Dependence 3
only if for Xl' ... , Xn distinct elements of E and ,11' ... , An in A the con
L
dition Aixi=O implies that A =A2=···=An=0.
1
i~n
Theorem 1. If E is a linearly independent set in L, then there is a vector
basis B of L such that B"ii?,E.
Proof. Let 6 be the set of all linearly independent subsets S of L
such that E~S; let Sl~S2 mean that Sl"ii?,S2. Then if60 isa simply
ordered subsystem of 6 and So is the union of all S in 60, So is also. a
linearly independent set; indeed, Xl' ... , Xn in So imply that there exist
Si in 6 with Xi in Si. Since 6 is simply ordered by inclusion, all Xi
0 0
belong to the largest Sj and are, therefore, linearly independent. Hence
SoE 6 and is an upper bound for 60. Zorn's lemma now applies to
assert that E is contained in a maximal element B of 6. This B is the
desired vector basis, for it is a linearly independent set and no linearly
independent set is larger.
Corollary 1. If Lo is a linear subspace of Land Bo is a vector basis for
Lo, then L has a vector basis B"ii?,Bo.
(6) If B = {xs: s E S} is a vector basis in L, each X in L has a repre
L L
sentation X= AsXs' where (J is a finite subset of S. If X= AsXs
L SEa SEUl
= JlsX" then As=Jls for all s in (Jl (l(J2 and As=O for all other s
SEa2
in (Jl and Jls=O for all other s in (J2. Hence each X=FO has a unique
representation in which all coefficients are non-zero, and 0 has no re
presentation in which any coefficient is non-zero. [Also see §2, (2c).]
This property characterizes bases among subsets of L.
Theorem 2. Any two vector bases Sand T of a linear space L have
the same cardinal number.
Proof. Symmetry of our assumptions and the Schroeder-Bernstein
theorem on comparability of cardinals (Kelley, p. 28) show that it
suffices to prove that S can be matched with a subset of T. Consider
the transitively ordered system of functions ([> consisting of those func
tions cP such that (a) the domain Dep~S and the range Rep~ T. (b) cP is
one-to-one between Dep and Rep. (c) Rep u (S\Dep) is a linearly indepen
dent set. Order ([> by: cP ~ cp' means that cp is an extension of cp'.
Every simply ordered subsystem ([>0 of ([> has an upper bound CPo:
Define Depo= U Dep and CPo(s)= cp(s) if sEDep and CPE([>o· This CPo
epetPo
is defined and is in ([>; it is an upper bound for ([>0. By Zorn's lemma
there is a maximal cp in ([>. We wish to show that Dep=S.
4 Chapter I. Linear Spaces
If not, then Rq> =!= T, for each s in the complement of Dq> is depen
dent on T but not on Rq>. If to is in T\Rq> , either to is linearly independent
of Rq> U (S\Dq» or is dependent on it. In the former case, for arbitrary
So in S\Dq> the extension cp' of cp for which cp' (so) = to has the properties
(a), (b), and (c), so cp is not maximal. In the latter case, by (c) and (6)
L L
to= Att+ IlsS'
tER.. s$D. .
where at least one Ilso is not zero, because to is independent of Rq>' If cp'
is the extension of cp for which cp'(so)=to, then cp' obviously satisfies
(a) and (b); also Rq>' U (S\Dq>') is linearly independent, because other
wise to would depend on Rq> U (S\Dq>')' a possibility prevented by the
choice of So, and again cp cannot be maximal.
This shows that if cp is maximal in ([>, then Dq>=S; then the cardinal
number of S is not greater than that of T. The Schroeder-Bernstein
theorem completes the proof of the theorem.
Definition 5. The cardinal number of a vector basis of L is called
the dimension of L.
The linear space with no element but 0 is the only linear space with
an empty vector basis; it is the unique linear space of dimension O.
(7) (a) If K is the complex field and if L is a vector space over K,
then L is also a vector space, which we shall call L(r)' over the real field R.
(b) The dimension of L(r) is twice that of L, for x and ix are linearly
independent in L(r)'
§ 2. Linear Functions and Conjugate Spaces
In this section again the nature of the field of scalars is unimportant
as long as it is not of characteristic 2.
Definition 1. If Land r. are linear spaces over the same field A,
a function F (sometimes to be called an operator) from L into r. is called
additive if F(x+y)=F(x)+F(y) for all x,y in L; homogeneous if
F(A x) = A F(x) for all A in A and x in L; linear if both additive and homo
geneous. A one-to-one linear F carrying L onto r. is an isomorphism
of Land r..
(1) (a) Let B be a vector basis of L and for each b in B let Yb be a
point of the linear space r.. Then there is a unique linear function F
from L into r. such that F(b) = Yb for all bin B; precisely, using §1, (6),
(L L
F Abb) = AbYb'
bEI1 bEI1
(b) If To is a linear function defined from a linear subspace Lo of L into
a linear space r., there is an extension T of To defined from L into r..