Fabian - Banach space theory (The basis for linear and nonlinear)

InChapter 1we present basic notions in Banach space theory and introduce theclassical Banach spaces, in particular sequence and function spaces... vi PrefaceIn Chapter 2we discuss two fu

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Canadian Mathematical Society Société mathématique du Canada

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Marián Fabian · Petr Habala · Petr Hájek ·

Banach Space Theory

The Basis for Linear and Nonlinear Analysis

123

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Marián Fabian

Mathematical Institute of the Academy

of Sciences of the Czech Republic

Mathematical Institute of the Academy

of Sciences of the Czech Republic

Žitná 25, Praha 1

11567 Prague, Czech Republic

hajek@math.cas.cz

Vicente Montesinos Universidad Politécnica de Valencia Departamento de Matematica Aplicada Camino de Vera s/n

46022 Valencia, Spain vmontesinos@mat.upv.es Václav Zizler

University of Alberta Department of Mathematical and Statistical Sciences Central Academic Building Edmonton T6G 2G1 Alberta, Canada zizler@math.cas.cz

Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2010938895

Mathematics Subject Classicication (2010):

Primary: 46Bxx

Secondary: 46A03, 46A20, 46A22, 46A25, 46A30, 46A32, 46A50, 46A55, 46B03, 46B04, 46B07, 46B10, 46B15, 46B20, 46B22, 46B25, 46B26, 46B28, 46B45, 46B50, 46B80, 46C05, 46C15, 46G05, 46G12, 47A10, 52A07, 52A21, 52A41, 58C20, 58C25

c

Springer Science+Business Media, LLC 2011

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Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

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In infinite dimensions, neighborhoods of points are not relatively compact, tinuous functions usually do not attain their extrema and linear operators are notautomatically continuous By introducing weak topologies, compactness can beobtained via Tychonoff’s theorem Similarly, functions often need to be perturbed sothat the problem of finding extrema is solvable To deal with problems in linear andnonlinear analysis, a good working knowledge of Banach space theory techniques

con-is needed It con-is the purpose of thcon-is introductory text to help the reader grasp the basicprinciples of Banach space theory and nonlinear geometric analysis

The text presents the basic principles and techniques that form the core of thetheory It is organized to help the reader proceed from the elementary part of thesubject to more recent developments This task is not easy Experience shows thatworking through a large number of exercises, provided with hints that direct thereader, is one of the most efficient ways to master the subject Exercises are ofseveral levels of difficulty, ranging from simple exercises to important results orexamples They illustrate delicate points in the theory and introduce the reader toadditional lines of research In this respect, they should be considered an integralpart of the text A list of remarks and open problems ends each chapter, presentingfurther developments and suggesting research paths

An effort has been made to ensure that the book can serve experts in relatedfields such as Optimization, Partial Differential Equations, Fixed Point Theory, RealAnalysis, Topology, and Applied Mathematics, among others

As prerequisites, basic undergraduate courses in calculus, linear algebra, andgeneral topology, should suffice

The text is divided into 17 chapters

InChapter 1we present basic notions in Banach space theory and introduce theclassical Banach spaces, in particular sequence and function spaces

v

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vi Preface

In Chapter 2we discuss two fundamental principles of Banach space theory,namely the Hahn–Banach Theorem on extension of bounded linear functionals andthe Banach Open mapping Theorem, together with some of their applications

InChapter 3we discuss weak topologies and their properties related to ness Then we prove the third fundamental principle, namely the Banach–SteinhausUniform Boundedness principle Special attention is devoted to weak compactness,

compact-in particular to the theorems of Eberlecompact-in, Šmulyan, Grothendieck and James, andthe theory of reflexive Banach spaces

InChapter 4we introduce Schauder bases in Banach spaces The possibility torepresent each element of the space as the sequence of its coefficients in a givenSchauder basis transfers the purely geometric techniques of the elementary Banachspace theory to the analytic computations of the classical analysis Although notevery separable Banach space admits a Schauder basis, the use of basic sequencesand Schauder bases with additional properties is one of the main tools in the inves-tigation of the structural properties of Banach spaces

InChapter 5we continue the study of the structure of Banach spaces by addingresults on extensions of operators, injectivity, and weak injectivity The core of thechapter is the theory of separable Banach spaces not containing isomorphic copies

of1

Chapter 6 is an introduction to some basic results in the geometry of dimensional Banach spaces and their connection to the structure of infinite-dimensional spaces We do not discuss the deeper parts of the theory, which essen-tially depend on measure theoretical techniques We introduce the notion of finiterepresentability, and prove the principle of local reflexivity We use the John ellip-soid to prove the Kadec–Snobar theorem and give a proof of Tzafriri’s theorem

finite-We indicate the connection of this result with Dvoretzky’s theorem Last part of thechapter is devoted to the Grothendieck inequality

InChapter 7we present an introduction to nonlinear analysis, namely to tional principles and differentiability

varia-In Chapter 8we study the interplay between differentiability of norms and thestructure of separable Asplund spaces

Chapter 9 introduces the subject of superreflexive spaces, whose structure isnicely described by the behavior of its finite-dimensional subspaces

Chapter 10studies the impact of the existence of higher order smooth norms onthe structure of the underlying space Special effort is devoted to countable compactspaces and pspaces

Chapter 11deals with the property of dentability and results on differentiation ofvector measures We prove some basic results on Banach spaces with the Radon–Nikodým property

Chapter 12introduces the reader to the nonlinear geometric analysis of Banachspaces Results on uniform and nonuniform homeomorphisms are presented, includ-ing Keller’s theorem and basic fixed points theorems (Brouwer, Schauder, etc) Wediscuss a proof of the homeomorphisms of Banach spaces and results on uniform,

in particular Lipschitz, homeomorphisms

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Preface vii

Chapter 13contains a basic study of an important class of non-separable Banachspaces, the weakly compactly generated spaces In particular, we discuss theirdecompositions and renormings We also study weakly compact operators, abso-lutely summing operators, and the Dunford–Pettis property

Chapter 14deals with results on weak topologies, focusing on special types ofcompacta (scattered, Eberlein, Corson, etc.)

Chapter 15 presents basic results in the spectral theory of operators We studycompact and self-adjoint operators

Chapter 16deals with the basic theory of tensor products We follow the Banachspace approach, focusing on the Grothendieck duality theory of tensor products,Schauder bases, applications to spaces of compact operators, etc We include Enflo’sexample of a Banach space without the approximation property

A short appendix (Chapter 17) has been included collecting some very basicdefinitions and results that are used in the text, for the reader’s immediate access

In writing the text we strived to avoid excessive technicalities, keeping each ject as elementary as reasonably possible Each chapter ends with a brief section ofRemarks and Open Questions, containing further known results and some problems

sub-in the area that are—to our best knowledge—open

Several more specialized books and survey articles appeared recently in Banachspace theory, as [AlKa], [BeLi], [BoVa], [CasGon], [DJT], [HMVZ], [JoLi3],[Kalt4], [KaKuLP], [LPT], [MOTV2], [Wojt], among others We hope that thepresent text can help both the student and the professional mathematician to getacquainted with the techniques needed in these directions We also made an effort

to make this text closer to a reference book in order to help researchers in Banachspace theory

We are grateful to many of our colleagues for suggestions, advice, and cussions on the subject of the book We thank our Institutions: the Institute ofMathematics of the Czech Academy of Sciences, the Czech Technical University

dis-in Prague, the Department of Mathematical and Statistical Sciences at the sity of Alberta, Edmonton, Canada, the Universidad Politécnica de Valencia, Spain,and its Instituto Universitario de Matemática Pura y Aplicada This work has beensupported by several Grant Agencies: The Czech National Grant Agency and theInstitutional Research Plan of the Academy of Sciences (Czech Republic), NSERCCanada, the Ministerio de Educación (Spain) and the Generalitat Valenciana (Valen-cia, Spain) The grants involved are IAA 100 190 610, IAA 100 190 901, GA ˇCR201/07/0394, No AVOZ 101 905 03 (Czech Republic), Proyecto MTM2008-03211(Spain), BEST/2009/096 (Generalitat Valenciana) and PR2009-0267 (Ministerio deEducación), NSERC-7926 (Canada)

Univer-We would like to thank the Springer Team for their interest in this project Inparticular, we are thankful to Keith F Taylor, Karl Dilcher, Mark Spencer, VaishaliDamle, and Charlene C Cerdas We thank also Eulalia Noguera for her help with thetex file, and to Integra Software Services Pvt Ltd, in particular Sankara Narayanan,for their assistance in editing the final version of this book

Above all, we are indebted to our families for their moral support and agement

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encour-viii Preface

We would be glad if this book inspired some young mathematicians to chooseBanach Space Theory and/or Nonlinear Geometric Analysis as their field of interest

We wish the reader a pleasant time spent over this book

Spring, 2010

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1 Basic Concepts in Banach Spaces 1

1.1 Basic Definitions 1

1.2 Hölder and Minkowski Inequalities, Classical Spaces C [0, 1], p , c0, L p [0, 1] 3

1.3 Operators, Quotients, Finite-Dimensional Spaces 13

1.4 Hilbert Spaces 24

1.5 Remarks and Open Problems 29

Exercises for Chapter 1 31

2 Hahn–Banach and Banach Open Mapping Theorems 53

2.1 Hahn–Banach Extension and Separation Theorems 54

2.2 Duals of Classical Spaces 60

2.3 Banach Open Mapping Theorem, Closed Graph Theorem, Dual Operators 65

3 Weak Topologies and Banach Spaces 83

3.1 Dual Pairs, Weak Topologies 83

3.2 Topological Vector Spaces 86

3.3 Locally Convex Spaces 94

3.4 Polarity 98

3.5 Topologies Compatible with a Dual Pair 100

3.6 Topologies of Subspaces and Quotients 103

3.7 Weak Compactness 104

3.8 Extreme Points, Krein–Milman Theorem 109

3.9 Representation and Compactness 112

3.10 The Space of Distributions 115

3.11 Banach Spaces 119

3.11.1 Banach–Steinhaus Theorem 119

3.11.2 Banach–Dieudonné Theorem 122

ix

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x Contents

3.11.3 The Bidual Space 125

3.11.4 The Completion of a Normed Space 126

3.11.5 Separability and Metrizability 127

3.11.6 Weak Compactness 129

3.11.7 Reflexivity 129

3.11.8 Boundaries 131

4 Schauder Bases 179

4.1 Projections and Complementability, Auerbach Bases 179

4.2 Basics on Schauder Bases 182

4.3 Shrinking and Boundedly Complete Bases, Perturbation 187

4.4 Block Bases, Bessaga–Pełczy´nski Selection Principle 194

4.5 Unconditional Bases 200

4.6 Bases in Classical Spaces 205

4.7 Subspaces of L pSpaces 213

4.8 Markushevich Bases 216

5 Structure of Banach Spaces 237

5.1 Extension of Operators and Lifting 237

5.2 Weak Injectivity 250

5.2.1 Schur Property 252

5.3 Rosenthal’s1Theorem 253

6 Finite-Dimensional Spaces 291

6.1 Finite Representability 291

6.2 Spreading Models 294

6.3 Complemented Subspaces in Spaces with an Unconditional Schauder Basis 298

6.4 The Complemented-Subspace Result 309

6.5 The John Ellipsoid 312

6.6 Kadec–Snobar Theorem 320

6.7 Grothendieck’s Inequality 323

6.8 Remarks 325

7 Optimization 331

7.1 Introduction 331

7.2 Subdifferentials: Šmulyan’s Lemma 336

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Contents xi

7.3 Ekeland Principle and Bishop–Phelps Theorem 351

7.4 Smooth Variational Principle 355

7.5 Norm-Attaining Operators 359

7.6 Michael’s Selection Theorem 361

8 C1-Smoothness in Separable Spaces 383

8.1 Smoothness and Renormings in Separable Spaces 383

8.2 Equivalence of Separable Asplund Spaces 385

8.3 Applications in Convexity 394

8.4 Smooth Approximation 402

8.5 Ranges of Smooth Maps 405

9 Superreflexive Spaces 429

9.1 Uniform Convexity and Uniform Smoothness, p and L pSpaces 429 9.2 Finite Representability, Superreflexivity 435

9.3 Applications 449

9.4 Remarks 453

10 Higher Order Smoothness 465

10.2 Smoothness in p 466

10.3 Countable James Boundary 468

11 Dentability and Differentiability 479

11.1 Dentability in X 479

11.2 Dentability in X∗ 486

11.3 The Radon–Nikodým Property 490

11.4 Extension of Rademacher’s Theorem 504

12 Basics in Nonlinear Geometric Analysis 521

12.1 Contractions and Nonexpansive Mappings 521

12.2 Brouwer and Schauder Theorems 526

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xii Contents

12.3 The Homeomorphisms of Convex Compact Sets: Keller’s

Theorem 533

12.3.1 Introduction 533

12.3.2 Elliptically Convex Sets 535

12.3.3 The Space T 537

12.3.4 Compact Elliptically Convex Subsets of2 538

12.3.5 Keller Theorem 541

12.3.6 Applications to Fixed Points 541

12.4 Homeomorphisms: Kadec’s Theorem 542

12.5 Lipschitz Homeomorphisms 545

13 Weakly Compactly Generated Spaces 575

13.2 Projectional Resolutions of the Identity 577

13.3 Consequences of the Existence of a Projectional Resolution 581

13.4 Renormings of Weakly Compactly Generated Banach Spaces 586

13.5 Weakly Compact Operators 591

13.6 Absolutely Summing Operators 592

13.7 The Dunford–Pettis Property 596

13.8 Applications 598

14 Topics in Weak Topologies on Banach Spaces 617

14.1 Eberlein Compact Spaces 617

14.2 Uniform Eberlein Compact Spaces 622

14.3 Scattered Compact Spaces 625

14.4 Weakly Lindelöf Spaces, Property C 629

14.5 Weak∗Topology of the Dual Unit Ball 634

15 Compact Operators on Banach Spaces 657

15.1 Compact Operators 657

15.2 Spectral Theory 661

15.3 Self-Adjoint Operators 668

16 Tensor Products 687

16.1 Tensor Products and Their Topologies 687

16.2 Duality of Injective Tensor Products 696

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Contents xiii

16.3 Approximation Property and Duality of Spaces of Operators 700

16.4 The Trace 708

16.5 Banach Spaces Without the Approximation Property 711

16.6 The Bounded Approximation Property 717

16.7 Schauder Bases in Tensor Products 721

17 Appendix 733

17.1 Basics in Topology 733

17.2 Nets and Filters 735

17.3 Nets and Filters in Topological Spaces 736

17.4 Ultraproducts 737

17.5 The Order Topology on the Ordinals 737

17.6 Continuity of Set-Valued Mappings 738

17.7 The Cantor Space 739

17.8 Baire’s Great Theorem 741

17.9 Polish Spaces 741

17.10 Uniform Spaces 741

17.11 Nets and Filters in Uniform Spaces 742

17.12 Partitions of Unity 743

17.13 Measure and Integral 744

17.13.1 Measure 744

17.13.2 Integral 745

17.14 Continued Fractions and the Representation of the Irrational Numbers 746

References 751

Symbol Index 777

Subject Index 781

Author Index 807

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Chapter 1

Basic Concepts in Banach Spaces

In this chapter we introduce basic notions and concepts in Banach space theory

As a rule we will work with real scalars, only in a few instances, e.g., in tral theory, we will use complex scalars.K denotes simultaneously the real (R) or

spec-complex (C) scalar field We use N for the set {1, 2, }.

All topologies are assumed to be Hausdorff, unless stated otherwise In particular,

by a compact space we mean a compact Hausdorff space By a neighborhood of a point x in a topological space T we mean any subset of T that contains an open subset O of T such that x ∈ O.

If(T, T ) is a topological space, and S is a nonempty subset, we shall write T

S

for the restriction of the topology T to S (and so (S, T ) becomes a topological

space) If there is no possibility of misunderstanding, the restricted topology will

be called again T For a brief review on basic topological notions see, e.g., the

Appendix

1.1 Basic Definitions

Definition 1.1 A non-negative function · on a vector (i.e., linear) space X is

called a norm on X if

(i)x ≥ 0 for every x ∈ X,

(ii)x = 0 if and only if x = 0,

(iii)λx = |λ| x for every x ∈ X and every scalar λ,

(iv)x + y ≤ x + y for every x, y ∈ X (the “triangle inequality”).

A vector space X with a norm · is denoted by (X, · ), and is called a normed

linear space (or just a normed space).

Note that the functionρ(x, y) := x − y, where x, y ∈ X, is indeed a metric

on X To check the triangle inequality we write

M Fabian et al., Banach Space Theory, CMS Books in Mathematics,

DOI 10.1007/978-1-4419-7515-7_1, C Springer Science+Business Media, LLC 2011

1

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2 1 Basic Concepts in Banach Spaces

All topological and uniform notions in normed spaces refer to the canonical ric given by the norm, unless stated otherwise In situations when more than onenormed space is considered, we will sometimes use · X to denote the norm of X

met-Definition 1.2 A Banach space is a normed linear space (X, · ) that is complete

in the canonical metric defined by ρ(x, y) = x − y for x, y ∈ X, i.e., every Cauchy sequence in X for the metric ρ converges to some point in X.

Let(X, · ) be a normed space The set B X := {x ∈ X : x ≤ 1} is said to be

the closed unit ball of X , and S X := {x ∈ X : x = 1} the unit sphere of (X, · ).

Given x0 ∈ X and r > 0, the set B(x0, r) := {x ∈ X : x − x0 ≤ r} is said to

be the closed ball centered at x0with radius r If M ⊂ X, then span(M) stands for

the linear hull—or span—of M, that is, the intersection of all linear subspaces of

X containing M Equivalently, span(M) is the smallest (in the sense of inclusion) linear subspace of X containing M, or the set of all finite linear combinations of elements in M Similarly, span(M) stands for the closed linear hull of M, i.e., the smallest closed linear subspace of X containing M.

If no misunderstanding can arise, by a “subspace” of a vector space we will mean

a linear subspace and, in case of normed spaces, a closed linear subspace

Definition 1.3 Let E be a vector space Given x , y ∈ E, the set [x, y] := {λx + (1 − λ)y : 0 ≤ λ ≤ 1} is called the closed segment defined by x and y If x = y, the set (x, y) := {λx + (1 − λ)y : 0 < λ < 1} is called the open segment defined

by x and y A subset C of a vector space E is called convex if [x, y] ⊂ C whenever

x, y ∈ C.

If M ⊂ X, the convex hull of M is the smallest convex subset of X containing

M, and will be denoted by conv(M); conv(M) denotes the closed convex hull of M, i.e., the smallest closed convex subset of X containing M.

Definition 1.4 Let U be a convex subset of a vector space V We say that a function

f : U → R is convex if fλx+(1−λ)y≤ λf (x)+(1−λ) f (y) for all x, y ∈ U and

λ ∈ [0, 1] We say that f is strictly convex if fλx+(1−λ)y< λf (x)+(1−λ) f (y) for all x, y ∈ U, x = y, and λ ∈ (0, 1).

For instance, every norm of a normed space X is a convex function on X Observe that a function f : U → R is convex if and only if the epigraph of f , i.e., the set

epi f := {(x, r) ∈ U × R : f (x) ≤ r} ⊂ X × R, is convex (the linear structure of

X× R is defined coordinatewise)

For subsets A , B of a vector space X and a scalar α we also write A + B :=

{a + b : a ∈ A, b ∈ B} and α A := {αa : a ∈ A}.

A set M ⊂ X is called symmetric if (−1)M ⊂ M, and balanced if αM ⊂ M for

allα ∈ K, |α| ≤ 1.

Let Y be a subspace of a normed space (X, · ) By (Y, · ) we denote Y

endowed with the restriction of · to Y if there is no risk of misunderstanding.

Fact 1.5 Let Y be a subspace of a Banach space X Then Y is a Banach space if

and only if Y is closed in X

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1.2 Hölder and Minkowski Inequalities, Classical Spaces 3

Proof: Assume that Y is closed Consider a Cauchy sequence {y n}∞

n=1in Y Since

the norm on Y is the restriction of the norm of X , the sequence is Cauchy in X and therefore converges to some y ∈ X As Y is closed, y ∈ Y and y n → y in Y

The other direction is proved by a similar argument

Definition 1.6 A subset M of a normed space (X, · ) is called bounded if there exists r > 0 such that M ⊂ r B X M is called totally bounded if for every ε > 0 the set M can be covered by a finite number of translates of εB X A sequence {x n } in X

is called bounded (totally bounded) if the set {x n : n ∈ N} is bounded (respectively,

totally bounded).

Note that every totally bounded set is already bounded See also Exercises1.47

and1.48for a description of total boundedness by usingε-nets,Definition 3.11, and

Section 17.10

1.2 Hölder and Minkowski Inequalities, Classical Spaces C [0, 1],

p, c0, Lp[0, 1]

We will now turn to some examples of Banach spaces

Definition 1.7 The symbol C [0, 1] denotes the vector space of all scalar valued

continuous functions on the interval [0, 1] (the vector addition and the scalar

mul-tiplication being defined pointwise), endowed with the norm

Cauchy sequence for every t ∈ [0, 1] Set f (t) := lim

n→∞ f n (t) This defines a scalar valued function f on [0, 1] It remains to show that f is continuous and f n → f

uniformly (i.e., in · ∞) Givenε > 0, there is n0such that| f n (t) − f m (t)| ≤ ε for every t ∈ [0, 1] and every n, m ≥ n0 By fixing n ≥ n0and letting m→ ∞ we get

| f n (t) − f (t)| ≤ ε for every n ≥ n0and every t ∈ [0, 1] Let t0∈ [0, 1] and ε > 0

be fixed Chooseδ > 0 so that | f n0(t) − f n0(t0)| < ε whenever |t − t0| < δ Then,

whenever|t − t0| < δ,

| f (t) − f (t0)| ≤ | f (t) − f n0(t)| + | f n0(t) − f n0(t0)| + | f n0(t0) − f (t0)| < 3ε Therefore f ∈ C[0, 1] It has been shown above that, for every n ≥ n0, f n−

f∞≤ ε This proves that f n − f ∞→ 0, so C[0, 1] is complete.

Analogously, the space C (K ) of continuous scalar functions on a compact space

K , endowed with the supremum norm, is a Banach space.

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We note that C [0, 1] is an infinite-dimensional Banach space To see this, it is

enough to produce, for any n ∈ N, a linearly independent set of n elements in

C [0, 1] The set of functions {1, t, t2, , t n−1} has this property More generally,

the space C (K ), where K is a compact topological space, is infinite-dimensional as soon as K is infinite; indeed, given a finite set of distinct points S := {k i : i =

1, 2, , n} in K , define the function δ k i on S for i = 1, 2, , n, where δ k is the

Kronecker delta function at k, i.e., δ k (k) = 1 and δ k (k ) = 0 for all k = k Extend

eachδ k i to a continuous function on K by using the Tietze–Urysohn theorem (see

Corollary 7.55) The resulting set of extended functions{δ k i : i = 1, 2, , n} is

linearly independent in C (K ).

Definition 1.9 The symbol n

∞ denotes the dimensional vector space of all

n-tuples of scalars (that is,Rn orCn ), the vector addition and the scalar multiplication being defined coordinatewise, endowed with the supremum norm · ∞defined for

x = (x1, , x n ) ∈ n

∞by

x∞= max{|x i | : i = 1, , n}.

Note that n

∞is a special case of a C(K ) space, where K := {1, , k}, endowed

with the discrete topology

In order to introduce the class of pspaces for 1< p < ∞ we need to prove the

following classical inequalities

Theorem 1.10 (Hölder inequality) Let p, q > 1 be such that 1

For p = 2, q = 2, the inequality (1.1) is known as the Cauchy–Schwarz inequality.

In the proof of Theorem1.10we will use the following statement

Lemma 1.11 Let p, q > 1 be such that 1

Proof: Consider the graph of the function y = x p−1, x ≥ 0, and the areas A1

of the region bounded by the curves y = x p−1, y = 0, x = a, and A2 of the

region bounded by the curves y = x p−1, x = 0, y = b (see Fig.1.1) Clearly,

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Fig 1.1 Two areas and a rectangle in the proof of Lemma1.11

which implies the desired inequality

Theorem 1.12 (Minkowski inequality) Let p ∈ [1, ∞) and n ∈ N Then for all

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Definition 1.13 Let p ∈ [1, ∞) The symbol n

p denotes the n-dimensional vector spaceKn , the vector addition and the scalar multiplication being defined coordinatewise, endowed with the norm defined for x = (x1, , x n ) ∈ n

By Minkowski’s inequality (1.2), · p is indeed a norm on X

The closed unit ball of2

1is the square with vertices±e1,±e2, where e i are the

standard unit vectors, e1= (1, 0) and e2= (0, 1) The unit ball of 2

∞is the square

with vertices(±e1± e2) The unit ball of 2

2is the disk of radius 1 centered at theorigin

Balls for other ps are somehow in between (see Fig.1.2)

3 / 2 B2

Fig 1.2 Several balls inR 2

The difference between n

1 and n

∞ becomes apparent once we increase the

dimension It is already apparent in three dimensions: The unit ball of3

∞is a cube,

whereas the unit ball of3

1is an octahedron The unit ball of3

2is a Euclidean ball(see Fig.1.3)

Fig 1.3 Several balls inR 3 B3 B3 B3

∞

Definition 1.14 Let p ∈ [1, ∞) The symbol p = p (N) denotes the vector space

of all scalar valued sequences x = {x i}∞

i=1 satisfying

|x i|p < ∞, the vector addition and the scalar multiplication being defined coordinatewise, endowed with the norm

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When a scalar sequence x = {x i}∞

i=1is considered an element of p, we use the

notation x = (x i ) This applies to other sequence spaces defined below.

To see that the definition is correct, we need to show that if x = (x i ), y = (y i ) ∈

p , then x + y ∈ pandx + y p ≤ x p + y p

For every n ≤ m ∈ N we have by the Minkowski inequality (1.2):

i→∞x i exists and is finite.

The symbol c0= c0(N) denotes the subspace of ∞consisting of all x = (x i ) such that lim

That the spaces p (N), ∞(N), c0(N), and c00(N) are infinite-dimensional

fol-lows from the fact that this is so for a vector space containing a linearly independent

set of n vectors for each n ∈ N Vectors {e i}n

i=1 are linearly independent, where

e i = (0, , 0, 1, 0, ), and 1 is at the ith position These vectors are called the

canonical unit vectors.

Proposition 1.16 (i) For p ∈ [1, ∞], the space is a Banach space.

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(ii) The spaces c and c0 are closed subspaces of ∞ and thus they are Banach spaces.

(iii) The space c00is not complete.

In the proof we will use the following lemma

Lemma 1.17 Let X be a normed space If a sequence {x n}∞n=1in X is Cauchy, then

it is bounded in X

Proof: Using the Cauchy property of{x n } we find n0∈ N such that x n − x n0 ≤ 1

for every n ≥ n0 Thenx n ≤ x n − x n0 + x n0 ≤ 1 + x n0 for every n ≥ n0

Therefore for every n∈ N:

x n ≤ max{x1, x2, , x n0 −1, x n0 + 1}.

Proof of Proposition1.16:

(i) It is easily checked, using a method similar to that for C [0, 1], that ∞ is a

Banach space Now consider p ∈ [1, ∞).

k=1converges to some x i for every i∈ N

Put x = (x i ) We will show that x ∈ p By Lemma1.17, there is a constant

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k→∞l kexists and is finite by showing that{l n}∞

k→∞l kexists and is finite.

Since x k → x = (x i ) in ∞, we have lim

k→∞x

k

i = x i for all i ∈ N We will show

that lim

i→∞x i = l, that is, x ∈ c Given ε > 0, we find n0so thatx k − x∞ < ε

and|l k − l| < ε for k ≥ n0, in particular|x k

i − x i|∞ < ε Then fix i0 so that

Thus lim x i = l and x ∈ c This shows that c is closed in ∞

Similarly we show that c0is a closed subspace of∞ By Fact1.5the spaces c and c0are Banach spaces

(iii) By Fact1.5, it is enough to show that c00 is not closed in c0 To this end,

consider x ∈ c0 defined by x = (x i ), where x i = 1

i for every i , and x n =

Let p ∈ [1, ∞) More generally, for an abstract nonempty set Γ we introduce

spaces p (Γ ) and c0(Γ ) The space p (Γ ) consists of all functions f : Γ → K

such that

γ ∈Γ | f (γ )| p < ∞, with the norm f p:=

γ ∈Γ | f (γ )| p 1

, wherethe sum is defined by

The space∞(Γ ) consists of all bounded functions f : Γ → K and is endowed

with the supremum norm · ∞ Its subspace c0(Γ ) consists of all functions f ∈

∞(Γ ) such that the set {γ ∈ Γ : | f (γ )| ≥ ε} is finite for every ε > 0 We also consider the space c00(Γ ) of all functions f ∈ ∞(Γ ) whose support supp( f ) :=

{γ ∈ Γ : f (γ ) = 0} is finite Note that c0(Γ ) = c00(Γ ) (closure in ∞(Γ )).

Similarly as above we show that p (Γ ) and c0(Γ ) are Banach spaces.

Note that every element f in c0(Γ ) has countable support Indeed, supp( f ) =

∞

n=1{γ : | f (γ )| ≥ 1/n} Since p (Γ ) ⊂ c0(Γ ) for all 1 ≤ p < ∞, the same is

true for every element in such an p (Γ ).

Definition 1.18 Let p ∈ [1, ∞) The symbol L p = L p [0, 1] denotes the vector

space of all classes of Lebesgue measurable scalar functions f defined almost everywhere on [0, 1] (we identify functions that are equal almost everywhere) such

that 01| f (t)| p dt < ∞, the vector addition and the scalar multiplication being defined pointwise, endowed with the norm

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 f p:= 1

0 | f (t)| p dt

1

Note that L pis a vector space Indeed,

so 01| f + g| p dt < ∞ and ( f + g) ∈ L p whenever f , g ∈ L p Similarly,α f ∈ L p

for everyα ∈ K and f ∈ L p These spaces are also infinite-dimensional; indeed,

given n ∈ N, the set {χ [i−1/n,i/n]}n

i=1, whereχ Sdenotes the characteristic function

of a set S, is linearly independent.

The triangle inequality for · p follows from the following versions of theHölder and Minkowski inequalities (Theorems1.19and1.20)

Theorem 1.19 (Hölder inequality) If p > 1, (1/p) + (1/q) = 1, f ∈ L p , and

1/p 1

0

|g(t)| q dt

1/q

(= f p g q ).

(1.4)

Proof: If f p = 0 or g q = 0, then the left-hand side of Equation (1.4) is also

zero, so the inequality holds Otherwise, put, for t ∈ [0, 1],

+1

q

|g(t)| q

g q q

The function f g is measurable and Equation (1.5) shows that its absolute value is

dominated by an integrable function, so f g is integrable, i.e., it is an element in L1.Integration of both members in inequality (1.5) gives (1.4)

Theorem 1.20 (Minkowski inequality) If p ≥ 1 and f, g ∈ L p , then

 f + g p ≤ f p + g p (1.6)

Proof: For p = 1 the assertion is trivial For p > 1 it follows from Hölder inequality

(Theorem1.19) Indeed, f + g ∈ L and| f + g| p−1∈ L , so

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Theorem 1.21 If p ∈ [1, ∞), then L p is a Banach space.

In the proof we will use the following lemma We say that a series∞

Lemma 1.22 A normed space X is a Banach space if and only if every absolutely

convergent series in X is convergent.

Proof: Assume that X is a Banach space and {x n } is a sequence in X such that

k+1x i < ε Therefore the sequence {s n}∞

n=1is Cauchy and thus convergent in

X So every absolutely convergent series in a Banach space is convergent.

Now assume that the condition on absolute convergence is true and let{x n}∞

n=1be

a Cauchy sequence in X First we show that {x n}∞

n=1has a convergent subsequence.

To this end, choose a subsequence n1< n2< such that x n k − x n l < 2 −kfor

l ≥ k This is done by induction: first choose n1such thatx n1−x l < 2−1for every

l ≥ n1 Then choose n2> n1such thatx n2 − x l < 2−2for every l ≥ n2, etc Put

x n0 = 0 and set y k = x n k − x n k−1 We have x n k =k

i=1y i, wherey k ≤ 21−kfor

every k ≥ 2 Asy k < ∞, by our assumption we get thaty kis convergent

and hence its partial sums x n k form a convergent sequence

To show that the whole sequence{x n}∞n=1converges to the same element is dard This concludes the proof of Lemma1.22

stan-For a slight improvement of Lemma1.22, see Exercise1.26

Proof of Theorem 1.21: By Lemma 1.22 we need to show that

n→∞g n (t) exists (finite or +∞) for every t ∈ [0, 1] Since g n ≥ 0, by Fatou’s

lemma, 01g p dt≤ lim inf

n→∞

1

0 g n p dt ≤ M p

Therefore g p ∈ L1[0, 1], so g(t) < ∞ on a set S of measure 1 in [0, 1] For

t ∈ S we have| f k (t)| = g(t) < ∞ Thus the sum s(t) := f k (t) is finite Define s(t) = +∞ for t /∈ S We need to show that s = f in L Denote

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s n (t) = n

k=1 f k (t) Then s = lim

n→∞s n almost everywhere (on S at least), so it

is a measurable function on[0, 1] We also have |s n (t)| ≤ g n (t) ≤ g(t) for every

t ∈ S Therefore |s(t)| ≤ g(t) almost everywhere on [0, 1], which implies that

almost everywhere on [0, 1], 2p g p dt < ∞, and |s n (t) − s(t)| → 0 almost

everywhere By the Lebesgue dominated convergence theorem (see, e.g., [Rudi2,Theorem 1.34]),

1

0

|s n (t) − s(t)| p

dt → 0.

This means thats n − s p→ 0, that is,f k = s in L p

Let f be a measurable function on [0, 1] We define

ess sup( f ) = infsup

f (t) : t ∈ N,

where the infimum is taken over all measurable subsets N of [0, 1] of Lebesgue

measure 1 One can also use other equivalent definitions, for instance

ess sup( f ) = infα : |{t ∈ [0, 1] : f (t) > α}| = 0,

where|A| denotes the Lebesgue measure of A Clearly f ≤ ess sup( f ) outside a

set of measure zero

Definition 1.23 The space L∞ = L∞[0, 1] denotes the vector space of all

classes of measurable functions f that are “essentially bounded,” i.e., such that

ess sup(| f |) < ∞, endowed with the norm f ∞:= ess sup(| f |).

Proposition 1.24 The space L∞is a Banach space.

Proof: It is easy to check that · ∞is a norm on the space L∞ We will show that

it is a Banach space First observe that { f n}∞

n=1being a Cauchy sequence in L∞

implies that lim

m ,n→∞ ( f n − f m ) = 0 uniformly on a set of measure 1 Indeed, for

k, l ∈ N put

Z k , = {t ∈ [0, 1] : | f k (t) − f l (t)| ≥ f k − f l∞}.

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On [0, 1]\

k ,l Z k ,l the functions{ f n} form a uniformly Cauchy and therefore

uni-formly convergent sequence From this and from the completeness of the supremum

norm, we obtain that L∞is a Banach space

The space L∞is infinite-dimensional; to see this follow the same argument as in

the L pcase

Given a measure space (Ω, μ) and p ∈ [1, ∞], the space L p (Ω, μ) (also denoted L p (μ)) can be introduced in a similar fashion (see, e.g., [DiUh] or [Wojt])

1.3 Operators, Quotients, Finite-Dimensional Spaces

We now begin an investigation of linear mappings between normed spaces Recall

that a mapping T from a vector space X over the field K into another vector space Y

overK is called linear if T (α1x1+ α2x2) = α1T (x1) + α2T (x2) for every α1, α2∈

K and x1, x2 ∈ X The vector space of all linear mapping from X into Y will be

denotedL(X, Y ) A linear functional on X is a linear mapping from X into the field

K Observe that a real linear functional (i.e., a linear functional from a vector space

over

between metric spaces is called Lipschitz, more precisely, C-Lipschitz, if there is

T (x), T (y) ≤ Cρ(x, y) for all x, y ∈ P In case of normed

spaces P, Q this inequality becomes T (x) − T (y) Q ≤ Cx − y P Note thatevery Lipschitz mapping is uniformly continuous

Proposition 1.25 Let (X, · X ) and (Y, · Y ) be normed spaces and let T be a linear mapping from X into Y The following are equivalent:

(i) T is continuous on X

(ii) T is continuous at the origin.

(iii) There is C > 0 such that T (x) Y ≤ Cx X for every x ∈ X.

ThereforeT (x) Y ≤ ε/δx X for every x ∈ X, which shows (iii) On the other

hand, (iii) clearly implies (ii) withδ := ε/C.

Assuming (iii) we obtain thatT (x) Y ≤ C whenever x ∈ B X , so T (B X ) is bounded On the other hand, if T (B X ) is bounded and, say, T (x) Y ≤ C for

every x ∈ B X , then for every x ∈ X, x = 0, we have T x

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Definition 1.26 Let X , Y be normed spaces A linear mapping from X into Y is called an operator An operator T from X into Y is called bounded if T (B X ) is bounded in Y

We define the operator norm of T by

Note thatT is the smallest number C that satisfies (iii) in Proposition1.25and if

T : X → Y is a bounded operator, then the kernel Ker(T ) := {x ∈ X : T (x) = 0}

is a closed subspace of X That B(X, Y ) is an infinite-dimensional space in the case that X is infinite-dimensional and Y is not reduced to {0} follows from the

Hahn–BanachTheorem 2.2, seeExercise 2.43

Proposition 1.27 Let X, Y be normed linear spaces If Y is a Banach space then

B(X, Y ) is also a Banach space.

Proof: The proof of completeness is similar to that for the space C [0, 1] In

particu-lar, if{T n } is a sequence of linear mappings from X into Y and T n (x) → T (x) for all x ∈ X, then T must be linear.

Operators in B(X, K) are called continuous linear functionals The operator

norm introduced in Definition1.26appears now as f = sup{| f (x)| : x ∈ B X}

for f ∈ B(X, K) We formally introduce the following definition.

Definition 1.28 Let (X, · ) be a normed space By X∗we denote the vector space

B(X, K) of all continuous linear functionals on X, endowed with the operator norm

 f = sup{| f (x)| : x ∈ B X }, called the canonical dual norm X∗is called the dual

space of X

Remark: If f ∈ X∗and x ∈ X, we shall use indistinctly the notation f (x) or f, x

for the action of f on x.

That the dual space does not reduce to{0} if X is different from {0} follows from

the Hahn–BanachTheorem 2.2 More precisely, if dim(X) = n for some n ∈ N,

then dim(X∗) = n, and if X is infinite-dimensional, so it is X∗, seeExercise 2.23.

Note that, by the definition, if f ∈ X∗and x ∈ X, then | f (x)| ≤ f · x The

canonical dual norm · is denoted sometimes by · ∗or · X∗ to emphasizethat we work in the dual space

SinceK (that is, R or C) is complete, using Proposition1.27we readily obtain

Proposition 1.29 X∗is a Banach space for every normed space X

A mappingϕ from a set D into a set R is called an injection (also called to-one) if ϕ(d ) = ϕ(d ) whenever d = d in D It is called a surjection (also

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one-1.3 Operators, Quotients, Finite-Dimensional Spaces 15

called onto) if for every r ∈ R there exists d ∈ D such that ϕ(d) = r The mapping

ϕ is called a bijection if it is both an injection and a surjection We use the terms injective, surjective, and bijective mapping, respectively.

An operator T ∈ B(X, Y ) is called a linear isomorphism from X onto Y , (or just

an isomorphism) if T is a bijection mapping from X to Y , and T−1∈ B(Y, X) If

T : X → Y is an isomorphism, we also say that T is invertible An isomorphism

from X onto X is called an automorphism Note that the inverse mapping of a linear bijection is also a linear mapping An operator T ∈ L(X, Y ) is an isomorphism

from X onto Y if and only if it is surjective and there exist constants C1and C2such

that, for all x ∈ X,

C1x ≤ T (x) ≤ C2x. (1.7)This follows from (iii) in Proposition1.25, and the fact that Equation (1.7) implies

that T is one-to-one Note (Exercise1.74) that (1.7) is equivalent to simultaneously

T ≤ C2andT−1 ≤ 1/C1 In geometrical terms, for an operator from X onto

Y , Equation (1.7) is equivalent to C1B Y ⊂ T (B X ) ⊂ C2B Y

Normed spaces X , Y are called linearly isomorphic (or just isomorphic) if there

is a linear isomorphism T from X onto Y It is easy to see that an isomorphism T

carries Cauchy (convergent) sequences onto Cauchy (convergent) sequences

There-fore, if X, Y are isomorphic normed spaces and X is a Banach space, then Y is a

Banach space as well

An operator T ∈ B(X, Y ) is called a linear isomorphism from X into Y (or

just an isomorphism from X into Y , alternatively an isomorphism into) if T is an isomorphism from X onto a subset T(X) of Y Clearly T (X) is a subspace of Y If

X is a Banach space then T (X) is complete too, so in particular it must be closed in

Y , by Fact1.5

An operator T ∈ B(X, Y ) is called a (linear) isometry from X into Y if

T (x) Y = x X for every x ∈ X Spaces X, Y are called (linearly) isometric

if there exists a linear isometry from X onto Y

Definition 1.30 Let X, Y be isomorphic normed spaces The Banach–Mazur

dis-tance between X and Y is defined by

d(X, Y ) = inf{T T−1 : T an isomorphism of X onto Y }.

Note that d(X, Y ) ≥ 1 and we have d(X, Z) ≤ d(X, Y ) d(Y, Z) The fact that

d(X, Y ) < d means that there is an isomorphism T of X onto Y such that 1

C1B Y ⊂

T (B X ) ⊂ C2B Y for some positive constants C1and C2satisfying C1C2< d Let a vector space X be endowed with two norms, denote X1 = (X, · 1) and X2 = (X, · 2) Norms · 1and · 2are called equivalent if the formal identity mapping I X : X → X defined by I X (x) = x is an isomorphism between the spaces X1 and X2, i.e., if there exist constants c, C > 0 such that cx2 ≤

x1≤ Cx2for every x ∈ X (see Equation (1.7)) According to Exercise1.74,

this is equivalent to c B1 ⊂ B2 ⊂ C B1, where B i denotes the closed unit ball of

X , i = 1, 2.

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We say that the space(X, ·2) is a renorming of the space (X, ·1) To renorm

a normed space means to endow the space with an equivalent norm

Definition 1.31 Let X and Y be normed spaces.

An operator T ∈ B(X, Y ) is called a compact operator if T (B X ) is compact in

Y The space of all compact operators from X into Y with the norm inherited from B(X, Y ) is denoted by K(X, Y ) If X = Y then we write K(X) instead of K(X, X).

An operator T ∈ B(X, Y ) is called a finite rank operator or a finite-dimensional

operator if dim

T (X) < ∞ By F(X, Y ) we denote the space of all finite rank operators from X into Y with the norm inherited from B(X, Y ) If X = Y then we write F(X) instead of F(X, X).

If f ∈ S X∗ and f does not attain its norm on B X , then f (B X ) = (−1, 1) (see

Exercise 3.161) Thus we have f ∈ K(X, R), yet f (B X ) is not compact Therefore, the closure T (B X ) in the definition of a compact operator cannot be dropped.

Example: A compact operator that is not a finite rank operator, but it is the limit

(in the operator norm) of a sequence of finite rank operators: Define T ∈ B(2)

by T (x) = (2 −i x

i ) for x = (x i ) Then T ∈ K(2)\F(2) Indeed, since

T (B 2) is a closed subset of the Hilbert cube (Exercise1.51), it is compact Thesequence of finite rank operators we are looking for is {T n }, where T n (x) := (2−1x

1, , 2 −n x

n , 0, 0, ), for all n ∈ N.

The identity operator on2is an example of a bounded non-compact operator,sincee i − e j =√2, if i = j.

Let T : X → Y be a finite rank operator Let n = dim T (X) A simple result in

Linear Algebra shows that we can find a biorthogonal system{e i ; g i}n

i=1in T (X) ×

T (X)∗ (i.e.,e i , g j = δ i , j for i, j ∈ {1, 2, , n}) In particular, {e i : i =

1, 2, , n} is a Hamel basis (i.e., an algebraic basis) of T (X) We define fi =

g i ◦ T ∈ X∗, i = 1, 2, , n Then for every x ∈ X we have T (x) =n

Definition 1.32 Let Z be a vector space A (linear) projection P on Z is a linear

mapping from Z into Z such that P ◦ P = P In this situation we say, too, that P

is a projection from Z onto P (Z) parallel to Ker P We shall often consider P as a mapping from the vector space Z onto the vector space P (Z).

Observe that associated to every linear projection on Z there exists an algebraic direct sum decomposition of Z into two subspaces X and Y , namely X = P Z and

Y = Ker P, in the sense that each z ∈ Z can be written in a unique way as z = x + y,

with x ∈ P Z and y ∈ Ker P Note that, in this case, x = Pz and y = z − Pz We

shall write in this case Z = X ⊕ Y , making sure that this is understood only in the

algebraic sense, with no topologies involved Conversely, given two subspaces X and Y of Z such that Z = X ⊕ Y (i.e., each z ∈ Z can be written in a unique way

as x + y, with x ∈ X and y ∈ Y ), then the mapping P : Z → Z defined as Pz = x

is a linear projection with range X and kernel Y

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More generally, given two vector spaces X and Y , the algebraic direct sum X ⊕Y

is the vector space of all ordered pairs (x, y), x ∈ X, y ∈ Y , with the vector operations defined coordinatewise The spaces X and Y are algebraically isomorphic

to the subspaces{(x, 0) : x ∈ X} and {(0, y) : y ∈ Y } of X ⊕ Y , respectively.

Definition 1.33 Let (X, · X ) and (Y, · Y ) be normed spaces The algebraic direct sum X ⊕ Y of X and Y becomes a normed space, called the topological

direct sum of X and Y and still denoted X ⊕ Y , when it is endowed with the norm

(x, y) := x X + y Y

The spaces X and Y are isometric to the subspaces {(x, 0) : x ∈ X} and {(0, y) :

y ∈ Y } of X ⊕ Y , respectively If X and Y are Banach spaces, then so is X ⊕ Y

Let Y be a closed subspace of a normed space X For x ∈ X we consider the

coset ˆx relative to Y ,

ˆx := {z ∈ X : (x − z) ∈ Y } = {x + y : y ∈ Y }.

The space X/Y := { ˆx : x ∈ X} of all cosets, together with the addition and scalar

multiplication defined by ˆx + ˆy = x +y and λ ˆx = λx, is clearly a vector space It is

easy to check that ˆx := inf{y : y ∈ ˆx} turns X/Y into a normed space Indeed,

for any z ∈ ˆx we have ˆx = inf{z − y : y ∈ Y } = dist(z, Y ) Therefore ˆx = ˆ0

if and only if x ∈ Y , as Y is closed.

If Y is a subspace of X , then dist (αx, Y ) = |α| dist(x, Y ) Therefore λ ˆx =

|λ| ˆx The triangle inequality follows since if x1, x2are in X and y1, y2are in Y ,

then

x1+ x2− (y1+ y2) ≤ x1− y1 + x2− y2.

Therefore dist(x1+ x2, Y ) ≤ dist(x1, Y ) + dist(x2, Y ).

Definition 1.34 Let Y be a closed subspace of a normed space X The space X /Y endowed with the canonical norm ˆx := inf{x : x ∈ ˆx}, where ˆx ∈ X/Y , is

called the quotient space of X with respect to Y

The canonical quotient mapping q : X → X/Y associates to every x ∈ X the coset

ˆx to which it belongs.

Obviously, the mapping q : X → X/Y is linear and continuous In fact, q ≤

1, as it follows from the definition of the norm in X /Y We will see later, as a

consequence of Lemma1.37, thatq = 1 if Y is a closed subspace of X such that

Y = X.

Proposition 1.35 Let Y be a closed subspace of a Banach space X Then X /Y is a Banach space.

Proof: Assume that

n ˆx n is an absolutely convergent series in X/Y For n ∈ N, choose x n ∈ ˆx nsuch thatx n ≤ ˆx n + 2−n Then

x nis an absolutely

conver-gent series in X that, according to Lemma1.22, converges Clearly, the canonical

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quotient mapping q : X → X/Y is continuous, so the seriesˆx n converges, too

It is enough to apply again Lemma1.22to conclude that X /Y is a Banach space.

It is easy to check that(X ⊕ Y )/ X is isomorphic to Y and (X ⊕ Y )/Y is phic to X However, if Y is a closed subspace of X , then X may not be isomorphic

isomor-to Y ⊕ (X/Y ), seeExercise 12.50

Proposition 1.36 Let X be a vector space If X is finite-dimensional, then any two

norms on X are equivalent.

In particular, all finite-dimensional normed spaces are Banach spaces and every normed space of dimension n is isomorphic to n

2 Consequently, if X is a Banach space and Y is a finite-dimensional subspace of

X , then Y is closed in X by Fact1.5

Proof: Let{e1, , e n } be an algebraic basis of X We introduce a new norm · 1

on X by x1 = |λ i | for x = λ i e i To check the triangle inequality, for

Thereforex − y ≤ x − y ≤ max{e i }x − y1

We note that S1 := {x ∈ X : x1 = 1} is compact in (X, · 1) Indeed, let

x1 < d for every non-zero x ∈ X From the latter

inequality we have c x1≤ x ≤ dx1for every x ∈ X, so · is equivalent to

· 1 Consequently, all norms are equivalent

If X is an n-dimensional vector space and T is a linear bijection from X onto n

2,

we can define a norm · 2on X by x2 = T (x) n Then T is an isometry of (X, · 2) onto n

2and · 2is an equivalent norm

We shall show later (see the paragraph after the proof ofTheorem 4.49) that nospace pis isomorphic to q for p = q.

Since (x, y) = (x X , y Y )1 for every(x, y) ∈ X ⊕ Y (see Definition

1.33), it follows from Proposition1.36 that |||(x, y)||| := (x p + y p )1 is an

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equivalent renorming of X ⊕ Y for every p ≥ 1 Such a renormed space is denoted

X ⊕ Yp

To characterize finite-dimensional normed spaces, we need the following

state-ment Recall that a subspace Y of a vector space X is called proper if Y = X.

Lemma 1.37 (Riesz) Let X be a normed space If Y is a proper closed subspace of

X then for every ε > 0 there is x ∈ S X such that dist (x, Y ) ≥ 1 − ε.

Proof: Choose an arbitrary elementˆz ∈ X/Y satisfying 1 > ˆz > 1 − ε Now pick

any z ∈ ˆz, z ≤ 1, and set x = z/z (see Fig.1.4) We get

dist(x, Y ) = dist(z, Y )/z = ˆz/z ≥ ˆz > 1 − ε

Fig 1.4 Riesz’s lemma

B X Y

x

1−ε

It follows from Lemma1.37thatq = 1, where q : X → X/Y is the canonical

quotient mapping

Theorem 1.38 Let X be a normed space The space X is finite-dimensional if and

only if the unit ball B X of X is compact.

Proof: If X is finite-dimensional, then B X is compact by the proof of tion1.36

Proposi-If X is infinite-dimensional, by Lemma 1.37 we find by induction an nite sequence {x n } in S X and such that dist

infi-x n , span{x1, , x n−1}) > 1

2 for

n = 2, 3, Thus dist(x n , x m ) > 1

2 for all n = m and therefore {x n} does not

have any convergent subsequence Hence B Xis not compact

Proposition 1.39 Every operator T from a finite-dimensional normed space X into

a normed space Y is continuous.

Proof: If dim X = n, then X is isomorphic to n

2(Proposition1.36) It is enough,

then, to prove that every operator T from n

2 into a normed space is continuous

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By linearity, T (x) =n

i=1x i T (e i ) For each i, |x i | ≤ x2, so the i th coordinate mapping x → x i from n

2intoK is continuous It follows that T is continuous.

A straightforward consequence of the previous result is that, for every n ∈ N,

two n-dimensional normed spaces X and Y are linearly isomorphic Indeed, if T :

X → Y is a linear isomorphism, then both T and T−1are continuous This result

follows from Proposition1.36, too

The following statement is an application of the preceding result to finite rankoperators A complementary result on the class of compact operators is included

Proposition 1.40 Let X , Y be normed spaces Then F(X, Y ) is a linear subspace

of K(X, Y ).

The space K(X, Y ) is a closed subspace of B(X, Y ); hence if Y is a Banach space, then K(X, Y ) is also a Banach space.

Proof: Since(T1+ T2)(X) ⊂ T1(X) + T2(X), F(X, Y ) is a subspace of B(X, Y ).

If T is a finite rank operator, then T (B X ) is a bounded set in a finite-dimensional (closed) space T (X), hence T (B X ) is compact (see the proof of Proposition1.36)

For operators T1, T2we also have

(αT1+ βT2)(B X ) ⊂ αT1(B X ) + βT2(B X ) ⊂ αT1(B X ) + βT2(B X ) and if T i are compact, i = 1, 2, the right hand side is a compact set (Exercise1.61).ThusK(X, Y ) is a subspace of B(X, Y ) We will show that it is closed there Consider T n ∈ K(X, Y ) such that lim T n = T in B(X, Y ) To show that T is

a compact operator, givenε > 0 we shall find a finite ε-net for T (B X ) (see

Exer-cises1.47and1.48) First note that T n → T in B(X, Y ) means that lim T n (x) =

T (x) uniformly for x ∈ B X Thus there exists n0such thatT n (x) − T (x) < ε/2 for x ∈ B X and n ≥ n0 Since T n0(B X ) is totally bounded in Y , there is a finite ε/2-net F for T n0(B X ) We claim that F is a finite ε-net for T (B X ) Indeed, given

x ∈ B X , we find y ∈ F such that T n0(x) − y < ε/2 Then T (x) − y ≤

T (x) − T n0(x) + T n0(x) − y < ε Therefore T is a compact operator The last

statement follows from this and Fact1.5

Remarks:

1 In general,F(X, Y ) is not a closed subspace of K(X, Y ) See the example after

Definition1.31

2 Note that if X is infinite-dimensional, then no isomorphism from X into Y can

be a compact operator by Theorem1.38 In particular, the identity operator I X

on an infinite-dimensional normed space X is never compact.

The spacesK(X, Y )∗andK(X, Y )∗∗are discussed inChapter 16.

Example: Let X = L2[0, 1] and K ∈ L2([0, 1]×[0, 1]) Define an operator T from

L [0, 1] into L [0, 1] by

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|K (s, t)|2

ds

1 0

Therefore if K is continuous on [0, 1] × [0, 1], then T (x) ∈ C[0, 1] In fact we

also showed that T (B L2) is a uniformly bounded (by M) and equicontinuous subset

of C [0, 1], hence by the Arzelà–Ascoli theorem, T (B L2) is relatively compact in

C [0, 1] Since the norm topology of L2[0, 1] is weaker than the topology of C[0, 1],

we have that T (B L2) is compact in L2[0, 1] and T is a compact operator.

If K ∈ L2([0, 1] × [0, 1]) then choose a sequence K nof continuous functions on

[0, 1] × [0, 1] such that

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A subset M of a topological space X is called dense in X if M, the closure of

M in X , is equal to X A subset M of a normed space X is called linearly dense if

span(M) = X

Remark: In general,F(X, Y ) is not a dense subspace of K(X, Y ) SeeChapter 16,

in particularTheorems 16.35and16.54

Definition 1.41 Let X be a topological space We say that X is separable if there

exists a countable dense subset of X

Proposition 1.42 (i) If p ∈ [1, ∞), then the space p is separable.

(ii) The spaces c and c0are separable.

(iii) The space ∞is not separable.

Proof: (i) Consider in pthe familyF formed by all finitely supported vectors with

rational coefficients ThenF is countable We will show that F is dense in p Given

x ∈ p andε > 0, choose n0 ∈ N such that∞

i =n0|x i|p

≤ ε p

2 and then find

rational numbers r1, r2, , r n0 −1such that|x i − r i|p≤ ε p

+ε p

2 < ε p

ThereforeF is dense in p

(ii) The proof of the separability of c0 is similar to (i) The case of the space

c needs only one adjustment, namely we define F as the family of vectors with

rational coefficients such that the vectors are eventually constant

(iii) Assume that ∞ is separable and let D be a dense countable set in ∞.The cardinal number of the family F of all subsets of N is the cardinality of the continuum c, in particular it is uncountable For every F ∈ F, let χ denote the

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characteristic function of F in N If F1, F2∈ F and F1 = F2, thenχ F1−χ F2∞≥

|χ F1(n) − χ F2(n)| = 1 for some n ∈ F1\F2or n ∈ F2\F1 For each F ∈ F, let

d F ∈ D be chosen such that χ F − d F < 1

4 If F1 = F2, thend F1 − d F2∞> 1

4.Indeed, if we hadd F1− d F2∞≤ 1

Similarly we prove that c0(Γ ) and p (Γ ), p ∈ [1, ∞], are nonseparable for Γ

uncountable

Proposition 1.43 (i) The space C [0, 1] is separable.

(ii) If p ∈ [1, ∞), then L p is separable.

(iii) The space L∞is not separable.

Proof: We only discuss the real case (real-valued functions) In the complex case,

we consider real and imaginary parts of functions separately in order to reduce theproblem to the real case

(i) The collectionP of polynomials on [0, 1] forms an algebra in C[0, 1] (i.e.,

a vector subspace closed for multiplication) that separates the points of[0, 1] (i.e.,

given s = t in [0, 1], there exists p ∈ P such that p(s) = p(t)) and contains

a constant function Therefore the closure P in C[0, 1] is C[0, 1] by the Stone–

Weierstrass theorem Since the countable set of polynomials on[0, 1] with rational

coefficients is dense inP, we obtain that C[0, 1] is separable.

(ii) It is proven in the theory of Lebesgue integral that C [0, 1] is dense in L p

(see, e.g., [Royd]) Therefore the real case of (ii) follows from (i)

(iii) Consider the functions f t := χ [0,t] , t ∈ [0, 1], where χ [0,t]is the istic function of[0, t] Then f t − f t  = 1 if t = t and a similar argument as in

character-the proof of Proposition1.42gives that L∞is not separable

Proposition 1.44 B(2) contains an isometric copy of ∞ and thus it is not separable.

Proof: Define a mappingϕ from ∞intoB(2) as follows: If (a i ) ∈ ∞, letϕ(a i )

be the bounded operator from2into2defined for(x i ) i ∈ 2by

ϕ(a i ): (x i ) i → (a i x i ) i

We claim thatϕ is a linear isometry from ∞intoB(2) It is enough to check that

if(a i )∞ = 1, then the operator ϕ(a i )has norm 1 in B(2) First note that if

x = (x i ) ∈ 2, then

(a i x i )2=|a i|2|x i|21

≤ (a i )∞|x i|21

= (a i )∞x2

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and thus the operatorϕ(a i )has norm at most 1 On the other hand, givenε > 0, choose n0such that|a n0| > 1 − ε Then ϕ(a i )(e n0) = a n0e n0, which has norm

|a n0| Letting ε → 0 we obtain ϕ(a i ) = 1

1.4 Hilbert Spaces

An inner product (or a scalar product or a dot product) on a vector space X is a

scalar valued function(·, ·) on X × X such that

(1) for every y ∈ X, the function x → (x, y) is linear,

(2)(x, y) = (y, x), where the bar denotes the complex conjugation,

(3)(x, x) ≥ 0 for every x ∈ X,

(4)(x, x) = 0 if and only if x = 0.

Note that by (1),(0, y) = 0 for any y ∈ X, hence also (y, 0) = 0 by (2).

Theorem 1.45 (Cauchy–Schwarz inequality) Let (x, y) be an inner product on a vector space X

(i) For x , y ∈ X we have |(x, y)| ≤√(x, x)√(y, y).

(ii) The function x :=√(x, x) is a norm on X.

Proof: (i) If(y, y) = 0, we have y = 0 and the inequality is satisfied Assume that (y, y) > 0 Then

0≤x−(x, y) (y, y) y , x − (x, y) (y, y) y

= (x, x) − |(x, y)| (y, y)2

and the statement follows

(ii) We will check the triangle inequality For x , y ∈ X we have

x + y2= (x + y, x + y) = (x, x) + (y, y) + (x, y) + (y, x)

= (x, x) + (y, y) + 2 Re(x, y) ≤ (x, x) + (y, y) + 2|(x, y)|

≤ (x, x) + (y, y) + 2(x, x)(y, y) =(x, x) +(y, y)2

=x + y2

.

One immediate consequence of Theorem1.45is that(·, ·) is a continuous

map-ping from(X, · ) × (X, · ) to the scalar field In particular, it implies that for a fixed vector y ∈ X, x → (x, y) is a continuous linear functional on X.

Definition 1.46 A Banach space (H, · ) is called a Hilbert space if there is an inner product (·, ·) on H such that x =√(x, x) for every x ∈ H.

It is straightforward to check that the norm · of a Hilbert space H satisfies the

parallelogram equality, namely, for every x , y ∈ H we have

x + y2+ x − y2= 2x2+ 2y2. (1.8)

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in the complex case.

On the other hand, assume that a norm · on a Banach space X satisfies the

parallelogram equality If we define(x, y) by the above equations, it turns out to be

an inner product (Exercise1.92) andx2= (x, x) Thus X is a Hilbert space.

Therefore a Banach space X is a Hilbert space if and only if every dimensional subspace of X is a Hilbert space The parallelogram equality (1.8) givesthat n

two-2,2, and L2are Hilbert spaces InChapter 4(see the paragraph after the proof

ofTheorem 4.53), we shall show that p and L pare not even isomorphic to a Hilbert

space for p = 2

Let H be a Hilbert space Let x , y ∈ H We say that x is orthogonal to y, denoted x ⊥ y, if (x, y) = 0 Let M ⊂ H We say that x is orthogonal to M,

denoted x ⊥ M, if x is orthogonal to every vector y from M.

Definition 1.47 Let F be a subspace of a Hilbert space H The set F⊥ := {h ∈

H : h ⊥ F} is called the orthogonal complement of F in H.

Fact 1.48 If F is a subspace of a Hilbert space H , then F⊥is a closed subspace

of H

Proof: Clearly, F⊥is a subspace If f, h, h n ∈ H, (h n , f ) = 0 and h n → h, then

(h, f ) = 0 This follows from the continuity of the linear functional h → (h, f )

discussed immediately before Definition1.46 Hence F⊥is a closed subspace.

Obviously F ∩ F⊥ = {0} Therefore every element z ∈ F + F⊥has a unique

expression in the form z = x + y with x ∈ F, y ∈ F⊥(i.e., F + F⊥is in fact an

algebraic direct sum) We can also see that the orthogonality gives

z2= (x + y, x + y) = x2+ y2. (1.11)

It follows that T : F ⊕ F⊥→ H defined by T (x, y) = x + y is an isomorphism of

F ⊕ F⊥onto F + F⊥⊂ H, so F + F⊥is a topological direct sum.

Theorem 1.49 (Riesz) Let F be a subspace of a Hilbert space H If F is closed,

then F + F⊥ = H Thus T : F ⊕ F⊥ → H defined by T (x, y) = x + y is an

isomorphism of F ⊕ F⊥onto H , and so H is the topological direct sum of F and

There-fore, if X, Y are isomorphic normed spaces and X is a Banach space, then Y is a

Banach space as well

An operator T ∈ B(X, Y ) is called a linear isomorphism... finite-dimensional normed spaces are Banach spaces and every normed space of dimension n is isomorphic to n

2 Consequently, if X is a Banach space and. .. (Exercise1.92) and< i>x2= (x, x) Thus X is a Hilbert space.

Therefore a Banach space X is a Hilbert space if and only if every dimensional subspace of X is a Hilbert space

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