PURE AND APPLIED MATHEMATICS

A Wiley-Interscience Series of Texts, Monographs, and Tracts

Founded by RICHARD COURANT
Editors: MYRON B. ALLEN III, DAVID A. COX, PETER LAX
Editors Emeriti: PETER HILTON, HARRY HOCHSTADT, JOHN TOLAND

A complete list of the titles in this series appears at the end of this volume.

Copyright © 2002 by John Wiley & Sons, Inc. All rights reserved.

Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data

Lax, Peter D.

      Functional analysis / Peter D. Lax.
         p. cm.
      Includes bibliographical references and index.
      ISBN 0-471-55604-1 (cloth : alk. paper)
      1. Functional analysis. I. Title.

      QA320.L345 2002

      515—dc21                                                             2001046547

20 19 18

CONTENTS

Foreword

1. Linear Spaces

2. Linear Maps

2.1 Algebra of linear maps

2.2. Index of a linear map

3. The Hahn-Banach Theorem

3.1 The extension theorem

3.2 Geometric Hahn-Banach theorem

3.3 Extensions of the Hahn-Banach theorem

4. Applications of the Hahn-Banach theorem

4.1 Extension of positive linear functionals

4.2 Banach limits

4.3 Finitely additive invariant set functions

5. Normed Linear Spaces

5.1 Norms

5.2 Noncompactness of the unit ball

5.3 Isometries

6. Hilbert Space

6.1 Scalar product

6.2 Closest point in a closed convex subset

6.3 Linear functionals

6.4 Linear span

7. Applications of Hilbert Space Results

7.1 Radon-Nikodym theorem

7.2 Dirichlet’s problem

8. Duals of Normed Linear Spaces

8.1 Bounded linear functionals

8.2 Extension of bounded linear functionals

8.3 Reflexive spaces

8.4 Support function of a set

9. Applications of Duality

9.1 Completeness of weighted powers

9.2 The Müntz approximation theorem

9.3 Runge’s theorem

9.4 Dual variational problems in function theory

9.5 Existence of Green’s function

10. Weak Convergence

10.1 Uniform boundedness of weakly convergent sequences

10.2 Weak sequential compactness

10.3 Weak* convergence

11. Applications of Weak Convergence

11.1 Approximation of the δ function by continuous functions

11.2 Divergence of Fourier series

11.3 Approximate quadrature

11.4 Weak and strong analyticity of vector-valued functions

11.5 Existence of solutions of partial differential equations

11.6 The representation of analytic functions with positive real part

12. The Weak and Weak* Topologies

13. Locally Convex Topologies and the Krein-Milman Theorem

13.1 Separation of points by linear functionals

13.2 The Krein-Milman theorem

13.3 The Stone-Weierstrass theorem

13.4 Choquet’s theorem

14. Examples of Convex Sets and Their Extreme Points

14.1 Positive functionals

14.2 Convex functions

14.3 Completely monotone functions

14.4 Theorems of Carathéodory and Bochner

14.5 A theorem of Krein

14.6 Positive harmonic functions

14.7 The Hamburger moment problem

14.8 G. Birkhoff’s conjecture

14.9 De Finetti’s theorem

14.10 Measure-preserving mappings

15. Bounded Linear Maps

15.1 Boundedness and continuity

15.2 Strong and weak topologies

15.3 Principle of uniform boundedness

15.4 Composition of bounded maps

15.5 The open mapping principle

16. Examples of Bounded Linear Maps

16.1 Boundedness of integral operators

16.2 The convexity theorem of Marcel Riesz

16.3 Examples of bounded integral operators

16.4 Solution operators for hyperbolic equations

16.5 Solution operator for the heat equation

16.6 Singular integral operators, pseudodifferential operators and Fourier integral operators

17. Banach Algebras and their Elementary Spectral Theory

17.1 Normed algebras

17.2 Functional calculus

18. Gelfand’s Theory of Commutative Banach Algebras

19. Applications of Gelfand’s Theory of Commutative Banach Algebras

19.1 The algebra C(S)

19.2 Gelfand compactification

19.3 Absolutely convergent Fourier series

19.4 Analytic functions in the closed unit disk

19.5 Analytic functions in the open unit disk

19.6 Wiener’s Tauberian theorem

19.7 Commutative images*-algebras

20. Examples of Operators and Their Spectra

20.1 Invertible maps

20.2 Shifts

20.3 Volterra integral operators

20.4 The Fourier transform

21. Compact Maps

21.1 Basic properties of compact maps

21.2 The spectral theory of compact maps

22. Examples of Compact Operators

22.1 Compactness criteria

22.2 Integral operators

22.3 The inverse of elliptic partial differential operators

22.4 Operators defined by parabolic equations

22.5 Almost orthogonal bases

23. Positive compact operators

23.1 The spectrum of compact positive operators

23.2 Stochastic integral operators

23.3 Inverse of a second order elliptic operator

24. Fredholm’s Theory of Integral Equations

24.1 The Fredholm determinant and the Fredholm resolvent

24.2 The multiplicative property of the Fredholm determinant

24.3 The Gelfand-Levitan-Marchenko equation and Dyson’s formula

25. Invariant Subspaces

25.1 Invariant subspaces of compact maps

25.2 Nested invariant subspaces

26. Harmonic Analysis on a Halfline

26.1 The Phragmén-Lindelöf principle for harmonic functions

26.2 An abstract Pragmén-Lindeöf principle

26.3 Asymptotic expansion

27. Index Theory

27.1 The Noether index

27.2 Toeplitz operators

27.3 Hankel operators

28. Compact Symmetric Operators in Hilbert Space

29. Examples of Compact Symmetric Operators

29.1 Convolution

29.2 The inverse of a differential operator

29.3 The inverse of partial differential operators

30. Trace Class and Trace Formula

30.1 Polar decomposition and singular values

30.2 Trace class, trace norm, and trace

30.3 The trace formula

30.4 The determinant

30.5 Examples and counterexamples of trace class operators

30.6 The Poisson summation formula

30.7 How to express the index of an operator as a difference of traces

30.8 The Hilbert-Schmidt class

30.9 Determinant and trace for operator in Banach spaces

31. Spectral Theory of Symmetric, Normal, and Unitary Operators

31.1 The spectrum of symmetric operators

31.2 Functional calculus for symmetric operators

31.3 Spectral resolution of symmetric operators

31.4 Absolutely continuous, singular, and point spectra

31.5 The spectral representation of symmetric operators

31.6 Spectral resolution of normal operators

31.7 Spectral resolution of unitary operators

32. Spectral Theory of Self-Adjoint Operators

32.1 Spectral resolution

32.2 Spectral resolution using the Cayley transform

32.3 A functional calculus for self-adjoint operators

33. Examples of Self-Adjoint Operators

33.1 The extension of unbounded symmetric operators

33.2 Examples of the extension of symmetric operators; deficiency indices

33.3 The Friedrichs extension

33.4 The Rellich perturbation theorem

33.5 The moment problem

34. Semigroups of Operators

34.1 Strongly continuous one-parameter semigroups

34.2 The generation of semigroups

34.3 The approximation of semigroups

34.4 Perturbation of semigroups

34.5 The spectral theory of semigroups

35. Groups of Unitary Operators

35.1 Stone’s theorem

35.2 Ergodic theory

35.3 The Koopman group

35.4 The wave equation

35.5 Translation representation

35.6 The Heisenberg commutation relation

36. Examples of Strongly Continuous Semigroups

36.1 Semigroups defined by parabolic equations

36.2 Semigroups defined by elliptic equations

36.3 Exponential decay of semigroups

36.4 The Lax-Phillips semigroup

36.5 The wave equation in the exterior of an obstacle

37. Scattering Theory

37.1 Perturbation theory

37.2 The wave operators

37.3 Existence of the wave operators

37.4 The invariance of wave operators

37.5 Potential scattering

37.6 The scattering operator

37.7 The Lax-Phillips scattering theory

37.8 The zeros of the scattering matrix

37.9 The automorphic wave equation

38. A Theorem of Beurling

38.1 The Hardy space

38.2 Beurling’s theorem

38.3 The Titchmarsh convolution theorem

Texts

A. Riesz-Kakutani representation theorem

A.1 Positive linear functionals

A.2 Volume

A.3 L as a space of functions

A.4 Measurable sets and measure

A.5 The Lebesgue measure and integral

B. Theory of distributions

B.1 Definitions and examples

B.2 Operations on distributions

B.3 Local properties of distributions

B.4 Applications to partial differential equations

B.5 The Fourier transform

B.6 Applications of the Fourier transform

B.7 Fourier series

C. Zorn’s Lemma

Author Index

Subject Index

FOREWORD

This book grew out of a course of lectures on functional analysis taught over many years to second-year graduate students at the Courant Institute of New York University. It is a graduate text, not a treatise or a monograph. Most of the chapters are short, for it is easier to digest material in small chunks. Not all topics can be presented briefly, so some of the chapters are longer. Theorems and lemmas, as well as equations, are numbered consecutively in each chapter.

The first 23 chapters make only a modest technical demand on the reader; this material would serve very well as text for an introductory graduate course on functional analysis. The rest of the material is well suited as text for a more advanced graduate course on functional analysis in general, or on Hilbert space in particular.

When I was a student, the only text on functional analysis was Banach’s original classic, written in 1932; Hille’s book appeared in time to serve as my graduation present. For Hilbert space there was Stone’s Colloquium publication, also from 1932, and Sz.-Nagy’s Ergebnisse volume. Since then, our cup hath run over; first came Riesz and Sz.-Nagy, then Dunford and Schwartz, Yosida, later Reed and Simon, and Rudin. For Hilbert space, there was Halmos’s elegant slender volume, and Achiezer and Glazman, all of which I read with pleasure and profit. Many, many more good texts have appeared since. Yet I believe that my book offers something new: the order in which the material is arranged, the interspersing of chapters on theory with chapters on applications, so that cold abstractions are made flesh and blood, and the inclusion of a very rich fare of mathematical problems that can be clarified and solved from the functional analytic point of view.

In choosing topics I heeded the warning of my teacher Friedrichs: “It is easy to write a book if you are willing to put into it everything you know about the subject.” I present the basic structure of the subject, and those more advanced topics that loom large in the body of mathematics. Among these are the spectral resolution and spectral representation of self-adjoint operators, the theory of compact operators, the Krein-Milman theorem, Gelfand’s theory of commutative Banach algebras, invariant subspaces, strongly continuous one-parameter semigroups. I discuss the index of operators, so important in calculating topological invariants; the celebrated trace formula of Lidskii, a powerful tool in analysis; the Fredholm determinant and its generalizations, rising again after almost a hundred years of hibernation; and scattering theory, another gift from physics to mathematics. I have also included some (but not all) special topics close to my heart.

What has been omitted? All of nonlinear functional analysis, for which I recommend the four-volume treatise by Zeidler. Operator algebras, except for Gelfand’s theory of commutative Banach algebras. I slight the geometric theory of Banach spaces; happily a handbook on this subject, edited by Bill Johnson and Joram Lindenstrauss is about to be published by North Holland.

What are the prerequisites? What every second-year graduate student-and many undergraduates—knows:

Naive set theory. Denumerable sets, the continuum, Zorn’s lemma.

Linear algebra. The alternative for linear maps, trace and determinant of a matrix, the spectral theory of general and symmetric matrices, functions of a matrix.

Point set topology. Complete metric spaces, the Baire category principle, Hausdorff spaces, compact sets, Tychonov’s theorem.

Basic theory of functions of a complex variable.

Real variables. The Arzela-Ascoli theorem, the Lebesgue decomposition of measures on images, Borel measure on compact spaces.

It is an accident of history that measure theory was invented before functional analysis. The usual presentations of measure theory fail to take advantage of the concepts and constructions of functional analysis. In an appendix on the Riesz-Kakutani representation theorem I show how to use the tools of functional analysis in measure theory. Another appendix summarizes the basic facts of Laurent Schwartz’s theory of distributions.

Many of the applications are to problems of partial differential equations. Here a nodding aquaintance with the Laplace and the wave equation would help, although an alert uninformed reader could pick up some of the basic results from these pages.

Like most mathematicians, I am no historian. Yet I have included historical remarks in some of the chapters, mainly where I had some firsthand knowledge, or where conventional history has been blatantly silent concerning the tragic fate of many of the founding fathers of functional analysis during the European horrors of the 1930s and 1940s.

I am indebted to many. I learned the rudiments of functional analysis, and how to apply them, from my teacher Friedrichs. Subsequently my views were shaped by the work of Tosio Kato, who has brought the power of functional analysis to bear on an astonishing range of problems. My happy and long collaboration with Ralph Phillips has led to some unusual uses of functional analysis. r learned much from Israel Gohberg, especially about the index of Toeplitz operators, from Bill Johnson about the fine points of the geometry of Banach spaces, and from Bob Phelps about Choquet’s theorem. I thank Reuben Hersh and Louise Raphael for their critique of the appendix on distributions, and Jerry Goldstein for his expert comments on the material on semigroups and scattering theory. To all of them, as well as to Gabor Francsics, my thanks.

Jerry Berkowitz and I alternated teaching functional analysis at the Courant Institute. This would be a better book had he lived and looked the manuscript over critically.

I thank Jeff Rosenbluth and Paul Chernoff for a careful reading of the early chapters, and Keisha Grady for TEXing the manuscript, and cheerfully making subsequent changes and corrections.

The lecture course was popular and successful with graduate students of the Courant Institute. I hope this printed version retains the spirit of the lectures.

PETER D. LAX

New York. NY
November 2001

1

LINEAR SPACES

A linear space X over a field F is a mathematical object in which two operations are defined: addition and multiplication by scalars.

Addition, denoted by +,as in

(1) images

is assumed to be commutative,

(2) images

associative,

(3) images

and to form a group, with the neutral element denoted as 0:

(4) images

The inverse of addition is denoted by-:

(5) images

The second operation is the multiplication of elements of X by elements k of the field F:

The result of this multiplication is again an element of X. Multiplication by elements of F is assumed to be associative,

(6) images

and distributive,

(7) images

as well as

(8) images

We assume that multiplication by the unit of F, denoted as 1, acts as the identity:

(9) images

These are the axioms of linear algebra. From them proceed to draw some deductions.

Set b = 0 in (8). It follows that for all x,

(10) images

Set a = 1, b = –1 in (8). Using (9) and (10), we deduce that for all x,

(11) images

The finite-dimensional linear spaces are dealt with in courses on linear algebra. In this book the emphasis is on the infinite-dimensional ones—those that are not finitedimensional. The field F will be either the real numbers images or the complex numbers images Here are some examples.

Example 1. X is the space of all polynomials in a single variable s, with real coefficients, here F = images

Example 2. X is the space of all polynomials in N variables s₁, … , s N, with real coefficients, here F = images

Example 3. G is a domain in the complex plane, and X the space of all functions complex analytic in G, here F = images.

Example 4. X = space of all vectors

with infinitely many real components, here F = images

Example 5. Q is a Hausdorff space, X the space of all continuous real-valued functions on Q, here F = images

Example 6. M is a C∞ differentiable manifold, X = C∞ (M), the space of all differentiable functions on M.

Example 7. Q is a measure space with measure m, X = L¹ (Q, m).

Example 8. X = LP(Q,m).

Example 9. X = harmonic functions in the upper half-plane.

Example 10. X = all solutions of a linear partial differential equation in a given domain.

Example 11. All meromorphic functions on a given Riemann surface; F = images.

We start the development of the theory by giving the basic constructions and concepts. Given two subsets S and T of a linear space X, we define their sum, denoted as S + T to be the set of all points x of the form x = y + z, y in S, z in T. The negative of a set S, denoted as –S, consists of all points x of the form x = –y, y in S.

Given two linear spaces Z and U over the same field, their direct sum is a linear space denoted as Z ⊕ U, consisting of ordered pairs (z,u), z in Z, u in U. Addition and multiplication by scalars is componentwise.

Definition. A subset Y of a linear space X is called a linear subspace of X if sums and scalar multiples of Y belong to Y.

Theorem 1.

(i) The sets {0} and X are linear subspaces of X.

(ii) The sum of any collection of subspaces is a subspace.

(iii) The intersection of any collection of subspaces is a subspace.

(iv) The union of a collection of subspaces totally ordered by inclusion is a sub-space.

Exercise 1. Prove theorem 1.

Let S be some subset of the linear space X. Consider the collection {Yσ } of all linear subspaces that contain the setS. This collection is not empty, since it certainly contains X.

Definition. The intersection ∩Yσ of all linear subspaces Yσ containing the set S is called the linear span of the set S.

Theorem2.

(i) The linear span of a set S is the smallest linear subspace containing S.

(ii) The linear span of S consists of all elements x of the form

(12) images

Proof Part (i) is merely a rephrasing of the definition of linear span. To prove part (ii), we remark that on the one hand, the elements of the form (12) form a linear subspace of X; on the other hand, every x of form (12) is contained in any subspace Y containing S.

REMARK 1. An element x of form (12) is called a linear combination of the points x₁, … ,xn. So theorem 1 can be restated as follows:

The linear span of a subset S of a linear space consists of all linear combinations of elements of S.

Definition. X a linear space, Y a linear subspace of X. Two points x₁ and x₂ of X are called equivalent modulo Y, denoted as x₁ = x₂ (mod Y), if x₁ − x₂ belongs to Y.

It follows from the properties of addition that equivalence mod Y is an equivalence relation, meaning that it is symmetric, reflexive, and transitive. That being the case, we can divide X into distinct equivalence classes mod Y. We denote the set of equivalence classes as X / Y. The set X / Y has a natural linear structure; the sum of two equivalence classes is defined by choosing arbitrary points in each equivalence class, adding them and forming the equivalence class of the sum. It is easy to check that the last equivalence class is independent of the representatives we picked; put differently, if x₁ ≡ z₁, x₂ ≡ z₂. then x₁ + x₂ ≡ z₁ + z₂ mod Y. Similarly we define multiplication by a scalar by picking arbitrary elements in the equivalence class. The resulting operation does not depend on the choice, since, if x₁ ≡ z₁, then kx₁ ≡ kz₁ mod Y.The quotient set X / Y endowed with this natural linear structure is called the quotient space of X mod Y. We define codim Y = dim X / Y.

Exercise 2. Verify the assertions made above.

As with all algebraic structures, so with linear structures we have the concept of isomorphism.

Definition. Two linear spaces X and Z over the same field are isomorphic if there is a one-to-one correspondence T carrying one into the other that maps sums into sums, scalar multiples into scalar multiples; that is,

(13) images

We define similarly homomorphism, called in this context a linear map.

Definition. X and U are linear spaces over the same field. A mapping M : X → U is called linear if it carries sums into sums, and scalar multiples into scalar multiples; that is, if for all x, y in X and all k in F

(14) images

X is called the domain of M, U its target.

REMARK 2. An isomorphism of linear spaces is a linear map that is one-to-one and onto.

Theorem 3.

(i) The image of a linear subspace Y of X under a linear map M : X → U is a linear subspace of U.

(ii) The inverse imaRe under M of a linear subspace V of U is a linear subspace of X.

Exercise 3. Prove theorem 3.

A very important concept in a linear space over the reals is convexity:

Definition. X is a linear space over the reals; a subset K of X is called convex if, whenever x and y belong to K, the whole segment with endpoints x, y, meaning all points of the form

(15) images

also belong to K.

Examples of convex sets in the plane are the circular disk, triangle, and semicircular disk. The following property of convex sets is an immediate consequence of the definition:

Theorem 4. Let K be a convex subset of a linear space X over the reals. Suppose that x₁, … , xn belong to K; then so does every x of the form

(16) images

Exercise 4. Prove theorem 4.

An x of form (16) is called a convex combination of x₁ ,x₂, … ,xn.

Theorem 5. Let X be a linear space over the reals.

(i) The empty set is convex.

(ii) A subset consisting of a single point is convex.

(iii) Every linear subspace of X is convex.

(iv) The sum of two convex subsets is convex.

(v) If K is convex, so is –K.

(vi) The intersection of an arbitrary collection of convex sets is convex.

(vii) Let {Kj} be a collection of convex subsets that is totally ordered by inclusion. Then their union ∪Kj is convex.

(viii) The image of a convex set under a linear map is convex.

(ix) The inverse image of a convex set under a linear map is convex.

Exercise 5. Prove theorem 5.

Definition. Let S be any subset of a linear space X over the reals. The convex hull of S is defined as the intersection of all convex sets containing S. The hull is denoted as images.

Theorem 6.

(i) The convex hull of S is the smallest convex set containing S.

(ii) The convex hull of S consists of all convex combinations (16) of points of S.

Exercise 6. Prove theorem 6.

Definition. A subset E of a convex set K is called an extreme subset of K if:

(i) E is convex and nonempty.

(ii) whenever a point x of E is expressed as

then both y and z belong to E.

An extreme subset consisting of a single point is called an extreme point of K.

Example 1. K is the interval 0 ≤ x ≤ 1; the two endpoints are extreme points.

Example 2. K is the closed disk

Every point on the circle x² + y₂ = 1 is an extreme point.

Example 3. The open disk

has no extreme points.

Example 4. K a polyhedron, including faces. Its extreme subsets are its faces, edges, vertices, and of course K itself.

Theorem 7. Let K be a convex set, E an extreme subset of K, and Fan extreme subset of E. Then F is an extreme subset of K.

Exercise 7. Prove theorem 7.

Theorem 8. Let M be a linear map of the linear space X into the linear space U. Let K be a convex subset of U, E an extreme subset of K. Then the inverse image of E is either empty or an extreme subset of the inverse image of K.

Exercise 8. Prove theorem 8.

Exercise 9. Give an example to show that the image of an extreme subset under a linear map need not be an extreme subset of the image.

Taking U to be one dimensional, we get

Corollary 8′. Denote by H a convex subset of a linear space X, ℓ a linear map of X into images, Hmin and Hmax the subsets of H, where ℓ achieves its minimum and maximum, respectively.

Assertion. When nonempty, Hmin and Hmax are extreme subsets of H.

2

LINEAR MAPS

2.1 ALGEBRA OF LINEAR MAPS

We recall from chapter 1 that a linear map from one linear space X into another, U, both over 1he same field of scalars, is a mapping of X into U,

that is an algebraic homomorphism:

(1) images

In this section we explore those properties of linear maps that depend on the purely algebraic properties (1), without any topological restrictions imposed on the spaces X,U.

The sum of two linear maps M and N of X into U, and the scalar multiple is defined as

(2) images

(3) images

This makes a linear space out of the set of linear maps of X into U. The space is denoted as images(X, U). Given two linear maps, one, M from X → U, the other, N from U → W, we can define their product as the composite map

(4) images

Since compositon of maps in general is associative, so is in particular the composition of linear maps. As we will see, composition is far from being commutative.

From now on we omit the bracket and denote the action of a linear map on x as

This notation suggests that the action of M on x is a kind of multiplication; indeed (1) and (2) give the distributive property of this kind of multiplication.

Exercise 1. Verify that the composite of two linear maps is linear, and that the distributive law holds:

Definition. A mapping is invertible if it maps X one-to-one and onto U.

If M is invertible, it has an inverse, denoted as M⁻¹, that satisfies

where I on the left is the identity mapping in X, on the right on U. If M is linear, so is M⁻¹.

Definition. The nullspace of M, denoted by NM, is the set of points mapped into zero.

The range of M, denoted by RM, is the image of X under M in U.

Theorem 1. Let M be a linear map of X → U.

(i) The nullspace NM is a linear subspace of X, the range RM a linear subspace of U.

(ii) M is invertible iff NM = {0} and RM = U.

(iii) M maps the quotient space X/NM one-to-one onto RM.

(iv) If M : X → U and K : U → W are both invertible, so is their product, and

(v) If KM is invertible, then

Exercise 2. Prove theorem 1.

We remark that when x = U = W are finite dimensional, then the invertibility of the product NM implies that N and M separately are invertible. This is not so in the infinite-dimensional case; take, for instance, X to be the space of infinite sequences

and define R and L to be right and left shift: Rx = (0, a₁, a₂, …), Lx = (a₂, a₃, …). Clearly, LR is the identity map, but neither R nor L are invertible; nor is RL the identity.

We formulate now a number of useful notions and results concerning mappings of a linear space into itself:

We denote by Nj the nullspace of the jth power of M:

(5) images

Theorem 2. The subspaces Nj defined in (5) have these properties:

(6) images

and

(7) images

Proof. Equation (6) is an immediate consequence of (5). To show (7), we claim that M maps Nj+1/Nj into Nj/Nj-1 in a one-to-one fashion. To see this, note that a nonzero element of Nj+1/Nj is represented by a point z in Nj+1 that does not lie in Nj. Clearly, Mz lies in Nj but not in Nj-1; this shows the one-to-oneness. It follows that Nj+1/Nj is isomorphic to a subspace of Nj/Nj-1, from which the statement (7) about dimension follows. When Nj+1/Nj is infinite-dimensional, so is Nj/Nj-1.

The following is an immediate corollary of equation (7):

Theorem 2′. Suppose that for some i the subspaces defined by (5) satisfy

(8) images

then

(8′) images

Definition. A subspace Y of X is called an invariant subspace of a linear map M: X → X if M maps Y into Y.

Theorem 3. Suppose that Y is an invariant subspace of X for a mapping M: X → X. Then

(i) there is a natural interpretation of M as a mapping X/Y → X/Y.

(ii) if both maps

are invertible, so is M: X→X.

Proof. We leave part (i) to the reader. In (ii) we show first that the null space of M on X is trivial. To see this, suppose that

then, since the nullspace of M on X/Y is assumed to be trivial, it follows that z belongs to Y. But since the nullspace of M on Y also is trivial, it follows that z = 0.

Next we show that M : X → X is onto, meaning that

(9) images

has a solution x₀ for every u₀ in X. To this end we solve equation (9) in two stages. First we solve the congruence

which is possible since M maps X/Y onto itself. Let x₁ be an element of the solution class; then x₁ satisfies

Therefore the solution x₀ of (9) is

where y is the solution in Y of

Such a solution exists since M is assumed to map Y onto Y.

We remark that whereas invertibility of M on Y and X/Y guarantees the invertibility of M on X, the converse by no means holds in spaces of infinite dimension. For example, Jet X be the space of all bounded continuous functions on images, S the shift operator

and Y the subspace of functions x(t) that vanish on the negative axis. Clearly, Y is shift invariant, and equally clearly, S is invertible on X, its inverse being the left unit shift. But S is not invertible on either Y or X/Y; on Y its range consists of functions x(t) that are zero for t ≤ 1,and on X/Y it has a nontrivial nullspace.

Exercise 3. What is the nullspace of S on X/Y?

The construction of invariant subspaces will be taken up in chapter 25. Here we gather the following useful observations:

Theorem 4. Let M he a linear map: X → X.

(i) For any y in X, the set {p(M)y}, where p represents any polynomial, is an invariant subspace of M.

(ii) Let T be a linear map: X → X that commllles with M: TM = MT. Then the nullspace of T is an invariant subspace of M.

Proof. Part (i) rests on the observation that if p(M) is a polynomial, so is Mp(M). Part (ii) follows from the observation that if M and T commute, and if z is in the nullspace of T : Tz = 0, then TMz = MTz = M0 = 0.

2.2 INDEX OF A LINEAR MAP

The next group of theorems describe an important special class of mappings.

Definition. A linear map G is called degenerate if its range is finite dimensional:

(10) images

Theorem 5. The degenerate maps form an ideal in the following sense:

(i) The sum of two degenerate maps is degenerate.

(ii) The product of a degenerate map with any linear map, in either order, is degenerate; that is, if G is degenerate, so are MG and GN, provided of course that the products can be defined.

Exercise 4. Prove theorem 5.

Definition. The linear maps M : X → U and L : U → X are pseudoinverse to each other if

(11) images

where I denotes the identity, G degenerate maps of X → X, and U → U, respectively.

Exercise 5. Prove that the right shift and the left shift described after theorem 1 are pseudoinverses of each other on the space of all sequences.

Theorem 6.

(i) if L and M are pseudoinverses of each other, so are L+G₁ and M+G₂, where G₁, G₂ are arbitrary degenerate maps.

(ii) Suppose that M : X → U and A: U → W have pseudoinverses L and B, respectively. Then AM and LB are pseudoinverse to each other.

Exercise 6. Prove theorem 6.

We recall the definition of codimension of a subspace R of a linear space U:

Theorem 7. A linear map M : X → U has a pseudoinverse if and only if

(12) images

Proof. For the "only if' part we use a lemma:

Lemma 8. If G is a degenerate map of X → X, then

(13) images

Proof. For x in NI+G,

This shows that

combined with (10) this shows the first part of (l3).

According to theorem 1 (iii), G maps X/NG one-to-one onto RG; so

(14) images

Obviously 1 + G maps every x in NG into itself; this shows that R1+G ⊃ NG. It follows from this relation that

(14′) images

Combining (14) and (14′), we conclude that codim R1+G ≤ dim RG: using (10), we deduce the second part of (13).

Suppose now that M has a pseudoinverse; then (11) holds. From the first relation in (11) we deduce that NM ⊂ NI+G and therefore dim NM ≤ dim NI+G; combining this with the first part of (13), we obtain the first part of (12). It follows from the second relation in (11) that RM ⊃ RI+G. Therefore

Combining this with the second relation in (13), we deduce the second part of (12).

For the “if” part we need:

Lemma 9. Every subspace N of a linear space has a complementary subspace Y, namely a linear subspace Y of X such that

meaning that every x in X can be decomposed uniquely as

(15) images

Proof. Consider all subspaces Y of X whose intersection with N is {0}, partially ordered by inclusion. Every totally ordered collection of Yj has as upper bound the union of the Yj. Zorn’s lemma shows that there is a maximal Y; this Y clearly has the property stated in the lemma. Now, if some x cannot be expressed of form (15), we could enlarge Y by adjoining x, contradicting the maximality of Y.

Note that the complementary subspace Y is in no way uniquely determined. Having determined a particular Y, we define the projection P onto N from the decomposition (15):

Exercise 7. Prove that P is a linear map.

Exercise 8. Show that when N has finite codimension, dim Y = codim N.

We return now to the proof of the “if” part of theorem 7: it follows from (15) that every equivalence class of X mod N contains exactly one element belonging to Y, and that this correspondence is an isomorphism:

Suppose that M : X → U satisfies conditions (12); we choose complementary subspaces Y and V for the nullspace and range of M:

(16) images

According to theorem 1 (iii), M maps X/NM one-to-one onto RM. Since X/NM is isomorphic with Y, we conclude that

is invertible. Denote its inverse by M⁻¹ and define the map K as follows:

(17) images

Using (16), we can extend K to all of U. Clearly,

(17′) images

We can rewrite (17′) as follows:

where P is projection onto N, Q projection onto V.It follows from this that K and M are pseudoinverse to each other in the sense of (11). Since P and Q are degenerate, the proof of theorem 7 is complete.

Definition. Let M : X → U be a linear map with a pseudoinverse. We define the index of such an M as

(18) images

It follows from theorem 7 that this definition makes sense.

Theorem 10. M : X → U and L : U → W are linear maps with pseudoinverse. Then the product LM has pseudoinverse, and

(19) images

Proof. By theorem 6 (ii), LM has a pseudoinverse. To prove (19), we want to use as a counting device the notion of an exact sequence:

Definition. A sequence of linear spaces V₀, V₁, …, Vn and a sequence of linear maps Tj : Vj → Vj+1

is called exact if the range of Tj is the nullspace of Tj+1.

Lemma 11. Suppose that all the Vj in the exact sequence above are finite dimensional and that

(20) images

Then

(20′) images

Proof. Decompose Vj as

where Nj is the nullspace of Tj and Yj complementary to Nj. The condition of exactness requires that Tj be an isomorphism of Yj with Nj+1 Since dim Vj = dim Nj + dim Yj. it follows that

(21) images

By (20),

(21′) images

Setting (21) and (21′) in the left side of (20′) shows that the alternating sum is zero.

To prove theorem 10, we construct the following exact sequence:

(22) images

The mapping I₀ identifies NM as a subspace of NLM. Q is the natural map of points of U into the equivalence classes of U mod RM containing them. E is the mapping of equivalence classes of W mod RLM into equivalence classes mod RL.

Exercise 9. Verify that (22) is an exact sequence.

We apply relation (20′) to the exact sequence (22), with

Using the definition of codimension, we can write (20′) as follows:

Using the definition (l8) of the index, we deduce the product formula (19) for the index.

The next result is called the stability of index:

Theorem 12. Let M : X → U be a linear map with a pseudoinverse, and G : X → U a degenerate linear map. Then M + G has a pseudoinverse, and

(23) images

Proof. We first verify (23) for U = X and M = I. For this we need a lemma:

Lemma 13. Let X be a linear space, and K : X → U a linear map of X into U that has a pseudoinverse. Let X₀ be a linear subspace of X that has finite codimension.

Then K₀ : X₀ → U, the restriction of K to X₀, has a pseudoinverse, and

(24) images

Proof. Factor K₀ as

(24′) images

where I₀ : X₀ → X is the identification map. Clearly NI₀ = {0}, RI₀ = X₀. so

(25) images

Now we apply the product formula (19) to (24′) and deduce (24).

Let G : X → X be a degenerate map; take K : X → X to be

(26) images

Clearly, I is a pseudoinverse to K. Take X₀ to be the nullspace of G:

(27) images

By (14), X₀ has finite codimension. Since G is zero on X₀, K₀, the restriction of K to X₀, is the identification map I₀. So by (25),

We apply now lemma 13 to K. By (24),

We deduce from the last two relations that

(28) images

for every K of form (26). This proves (23) for M = I.

We take now M as any map with a pseudoinverse; denote by L : U → X a pseudoinverse of M. By definition,

G′ degenerate. So by (28),

(29) images

Using the product formula (19), we get from (29) that

(30) images

As we saw in theorem 6 (i), for degenerate G, L is also a pseudoinverse of M + G. Therefore. using (30), once more we deduce that

(30′) images

Combining (30) and (30′), we get (23).

Notes

The first part of this chapter is standard fare. The nonstandard items are as follows:

(i) The notion of the index of linear maps that have a pseudoinverse, theorem 7.

(ii) The product formula for the index, theorem 10.

(iii) The invariance of the index under perturbation by degenerate maps, theorem 12.

Strange to say, these results of linear algebra were first discovered in the setting of bounded maps of normed linear spaces. That they hold without any topological assumptions has remained a folk theorem. The first statement and proof of the multiplicative property in print is due to Donald Sarason. The proof presented here, using exact sequences, is due to Sergiu Klainerman.

BIBLIOGRAPHY

Sarason, D. The multiplication theorem for Fredholm operatms. Am. Math. Monthly, 94 (1987): 68–70.

3

THE HAHN-BANACH THEOREM

3.1 THE EXTENSION THEOREM

The result named in the title of this chapter is remarkable for its simplicity and for its far-reaching consequences. It deals with the extension of linear functionals.

Definition. A linear functional ℓ is a mapping of a linear space X over a field F into F, that is linear:

for all x, y in X and

for all k in F.

In this section we will mainly deal with linear spaces over the field of reals, and real number valued linear functionals.

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