Unformatted text preview: FUNCTIONAL ANALYSIS
Theo B¨
uhler
ETH Z¨
urich Dietmar A. Salamon
ETH Z¨
urich
8 June 2017 ii Preface
These are notes for the lecture course “Functional Analysis I” held by the
second author at ETH Z¨
urich in the fall semester 2015. Prerequisites are
the first year courses on Analysis and Linear Algebra, and the second year
courses on Complex Analysis, Topology, and Measure and Integration.
The material of Subsection 1.3.3 on elementary Hilbert space theory, Subsection 5.4.2 on the Stone–Weierstraß Theorem, and the appendices on the
Lemma of Zorn and Tychonoff’s Theorem has not been covered in the lectures. These topics were assumed to have been covered in previous lecture
courses. They are included here for completeness of the exposition.
The material of Subsection 2.4.4 on the James space, Section 5.5 on the
functional calculus for bounded normal operators, Chapter 6 on unbounded
linear operators, Subsection 7.3.2 on Banach space valued Lp functions, Subsection 7.3.4 on self-adjoint and unitary semigroups, and Section 7.4 on analytic semigroups was not part of the lecture course (with the exception of
some of the basic definitions in Chapter 6 that are relevant for infinitesimal
generators of strongly continuous semigroups, namely, parts of Section 6.2 on
the dual of an unbounded operator on a Banach space and Subsection 6.3.1
on the adjoint of an unbounded operator on a Hilbert space).
7 June 2017 Theo B¨
uhler
Dietmar A. Salamon iii iv Contents
Introduction 1 1 Foundations
1.1 Metric Spaces and Compact Sets . . . . . . . . . . . .
1.1.1 Banach Spaces . . . . . . . . . . . . . . . . . .
1.1.2 Compact Sets . . . . . . . . . . . . . . . . . . .
1.1.3 The Arzel`a–Ascoli Theorem . . . . . . . . . . .
1.2 Finite-Dimensional Banach Spaces . . . . . . . . . . . .
1.2.1 Bounded Linear Operators . . . . . . . . . . . .
1.2.2 Finite-Dimensional Normed Vector Spaces . . .
1.2.3 Quotient and Product Spaces . . . . . . . . . .
1.3 The Dual Space . . . . . . . . . . . . . . . . . . . . . .
1.3.1 The Banach Space of Bounded Linear Operators
1.3.2 Examples of Dual Spaces . . . . . . . . . . . . .
1.3.3 Hilbert Spaces . . . . . . . . . . . . . . . . . . .
1.4 Banach Algebras . . . . . . . . . . . . . . . . . . . . .
1.5 The Baire Category Theorem . . . . . . . . . . . . . .
1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
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52 2 Principles of Functional Analysis
2.1 Uniform Boundedness . . . . . . . .
2.2 Open Mappings and Closed Graphs .
2.2.1 The Open Mapping Theorem
2.2.2 The Closed Graph Theorem .
2.2.3 Closeable Operators . . . . .
2.3 Hahn–Banach and Convexity . . . .
2.3.1 The Hahn–Banach Theorem .
2.3.2 Positive Linear Functionals . . .
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. vi CONTENTS 2.4 2.5 2.3.3 Separation of Convex Sets . . . .
2.3.4 The Closure of a Linear Subspace
2.3.5 Complemented Subspaces . . . .
2.3.6 Orthonormal Bases . . . . . . . .
Reflexive Banach Spaces . . . . . . . . .
2.4.1 The Bidual Space . . . . . . . . .
2.4.2 Reflexive Banach Spaces . . . . .
2.4.3 Separable Banach Spaces . . . . .
2.4.4 The James Space . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . .
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. 3 The Weak and Weak* Topologies
3.1 Topological Vector Spaces . . . . . . . . . . . . . . .
3.1.1 Definition and Examples . . . . . . . . . . . .
3.1.2 Convex Sets . . . . . . . . . . . . . . . . . . .
3.1.3 Elementary Properties of the Weak Topology
3.1.4 Elementary Properties of the Weak* Topology
3.2 The Banach–Alaoglu Theorem . . . . . . . . . . . . .
3.2.1 The Separable Case . . . . . . . . . . . . . . .
3.2.2 Invariant Measures . . . . . . . . . . . . . . .
3.2.3 The General Case . . . . . . . . . . . . . . . .
3.3 The Banach–Dieudonn´e Theorem . . . . . . . . . . .
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3.4 The Eberlein–Smulyan
Theorem . . . . . . . . . . . .
3.5 The Kreˇın–Milman Theorem . . . . . . . . . . . . . .
3.6 Ergodic Theory . . . . . . . . . . . . . . . . . . . . .
3.6.1 Ergodic Measures . . . . . . . . . . . . . . . .
3.6.2 Space and Times Averages . . . . . . . . . . .
3.6.3 An Abstract Ergodic Theorem . . . . . . . . .
3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . .
4 Fredholm Theory
4.1 The Dual Operator . . . . . . . . .
4.1.1 Definition and Examples . .
4.1.2 Duality . . . . . . . . . . .
4.1.3 The Closed Image Theorem
4.2 Compact Operators . . . . . . . . .
4.3 Fredholm Operators . . . . . . . .
4.4 Composition and Stability . . . . . .
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. 195 CONTENTS
4.5 vii Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 5 Spectral Theory
5.1 Complex Banach Spaces . . . . . . . . . . . . . . . .
5.1.1 Definition and Examples . . . . . . . . . . . .
5.1.2 Integration . . . . . . . . . . . . . . . . . . .
5.1.3 Holomorphic Functions . . . . . . . . . . . . .
5.2 The Spectrum . . . . . . . . . . . . . . . . . . . . . .
5.2.1 The Spectrum of a Bounded Linear Operator
5.2.2 The Spectral Radius . . . . . . . . . . . . . .
5.2.3 The Spectrum of a Compact Operator . . . .
5.2.4 Holomorphic Functional Calculus . . . . . . .
5.3 Operators on Hilbert Spaces . . . . . . . . . . . . . .
5.3.1 Complex Hilbert Spaces . . . . . . . . . . . .
5.3.2 The Adjoint Operator . . . . . . . . . . . . .
5.3.3 The Spectrum of a Normal Operator . . . . .
5.3.4 The Spectrum of a Self-Adjoint Operator . . .
5.4 The Spectral Mapping Theorem . . . . . . . . . . . .
5.4.1 C* Algebras . . . . . . . . . . . . . . . . . . .
5.4.2 The Stone–Weierstraß Theorem . . . . . . . .
5.4.3 Functional Calculus for Self-Adjoint Operators
5.5 Spectral Representations . . . . . . . . . . . . . . . .
5.5.1 The Gelfand Representation . . . . . . . . . .
5.5.2 C* Algebras of Normal Operators . . . . . . .
5.5.3 Functional Calculus for Normal Operators . .
5.6 Spectral Measures . . . . . . . . . . . . . . . . . . . .
5.6.1 Projection Valued Measures . . . . . . . . . .
5.6.2 Measurable Functional Calculus . . . . . . . .
5.7 Cyclic Vectors . . . . . . . . . . . . . . . . . . . . . .
5.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . .
6 Unbounded Operators
6.1 Unbounded Operators on Banach Spaces . . . . .
6.1.1 Definition and Examples . . . . . . . . . .
6.1.2 The Spectrum of an Unbounded Operator
6.1.3 Spectral Projections . . . . . . . . . . . .
6.2 The Dual of an Unbounded Operator . . . . . . .
6.3 Unbounded Operators on Hilbert Spaces . . . . . .
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. 323 viii CONTENTS 6.4
6.5
6.6 6.3.1 The Adjoint of an Unbounded Operator
6.3.2 Unbounded Self-Adjoint Operators . . .
6.3.3 Unbounded Normal Operators . . . . . .
Functional Calculus . . . . . . . . . . . . . . . .
Spectral Measures . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . .
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. 7 Semigroups of Operators
7.1 Strongly Continuous Semigroups . . . . . . . . . . .
7.1.1 Definition and Examples . . . . . . . . . . .
7.1.2 Basic Properties . . . . . . . . . . . . . . . .
7.1.3 The Infinitesimal Generator . . . . . . . . .
7.2 The Hille–Yosida–Phillips Theorem . . . . . . . . .
7.2.1 Well-Posed Cauchy Problems . . . . . . . .
7.2.2 The Hille–Yosida–Phillips Theorem . . . . .
7.2.3 Contraction Semigroups . . . . . . . . . . .
7.3 Semigroups and Duality . . . . . . . . . . . . . . .
7.3.1 Banach Space Valued Measurable Functions
7.3.2 The Banach Space Lp (I, X) . . . . . . . . .
7.3.3 The Dual Semigroup . . . . . . . . . . . . .
7.3.4 Semigroups on Hilbert Spaces . . . . . . . .
7.4 Analytic Semigroups . . . . . . . . . . . . . . . . .
7.4.1 Properties of Analytic Semigroups . . . . . .
7.4.2 Generators of Analytic Semigroups . . . . .
7.4.3 Examples of Analytic Semigroups . . . . . .
7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . .
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. 419 A The Lemma of Zorn 421 B Tychonoff ’s Theorem 427 References 431 Notation 435 Index 437 Introduction
Classically, functional analysis is the study of function spaces and linear operators between them. The relevant function spaces are often equipped with
the structure of a Banach space and many of the central results remain valid
in the more general setting of bounded linear operators between Banach
spaces or normed vector spaces, where the specific properties of the concrete
function space in question only play a minor role. Thus, in the modern guise,
functional analysis is the study of Banach spaces and bounded linear operators between them, and this is the viewpoint taken in the present manuscript.
This area of mathematics has both an intrinsic beauty, which we hope to
convey to the reader, and a vast number of applications in many fields of
mathematics. These include the analysis of PDEs, differential topology and
geometry, symplectic topology, quantum mechanics, probability theory, geometric group theory, dynamical systems, ergodic theory, and approximation
theory, among many others. While we say little about specific applications,
they do motivate the choice of topics covered in this book, and our goal is
to give a self-contained exposition of the necessary background in abstract
functional analysis for many of the relevant applications.
The manuscript is addressed primarily to third year students of mathematics or physics, and the reader is assumed to be familiar with first year
analysis and linear algebra, as well as complex analysis and the basics of point
set topology and measure and integration. For example, this manuscript does
not include a proof of completeness and duality for Lp spaces.
There are naturally many topics that go beyond the scope of the present
manuscript, such as Sobolev spaces and PDEs, which would require a book on
its own and, in fact, very many books have been written about this subject;
here we just refer the interested reader to [11, 15, 16]. We also restrict the
discussion to linear operators and say nothing about nonlinear functional
analysis. Other topics not covered include the Fourier transform (see [2, 32,
1 2 CONTENTS 54]), maximal regularity for semigroups (see [51]), the space of Fredholm
operators on an infinite-dimensional Hilbert space as a classifying space for
K-theory (see [5, 6, 7, 28]), Quillen’s determinant line bundle over the space of
Fredholm operators (see [46, 52]), and the work of Gowers [17] and Argyros–
Haydon [4] on Banach spaces on which every bounded linear operator is the
sum of scalar multiple of the identity and a compact operator. Here is a
description of the content of the book, chapter by chapter.
Chapter 1 discusses some basic concepts that play a central role in the
subject. It begins with a section on metric spaces and compact sets which
includes a proof of the Arzel`a–Ascoli theorem. It then moves on to establish
some basic properties of finite-dimensional normed vector space spaces and
shows, in particular, that a normed vector space is finite-dimensional if and
only if the unit ball is compact. The first chapter also introduces the dual
space of a normed vector space, explains several important examples, and
contains an introduction to elementary Hilbert space theory. It then introduces Banach algebras and shows that the group of invertible elements is an
open set. It closes with a proof of the Baire category theorem.
Chapter 2 is devoted to the three fundamental principles of functional
analysis. They are the Uniform Boundedness Principle (a pointwise bounded
family of bounded linear operators on a Banach space is bounded), the Open
Mapping Theorem (a surjective bounded linear operator between Banach
spaces is open), and the Hahn–Banach Theorem (a bounded linear functional on a linear subspace of a normed vector space extends to a bounded
linear functional on the entire normed vector space). An equivalent formulation of the Open Mapping Theorem is the Closed Graph Theorem (a linear
operator between Banach spaces is bounded if and only if it has a closed
graph) and a corollary is the Inverse Operator Theorem (a bijective bounded
linear operator between Banach spaces has a bounded inverse). A slightly
stronger version of the Hahn–Banach theorem, with the norm replaced by
a quasi-seminorm, can be reformulated as the geometric assertion that two
convex subsets of a normed vector space can be separated by a hyperplane
whenever one of them has nonempty interior. The chapter also discusses
reflexive Banach spaces and includes an exposition of the James space.
The subject of Chapter 3 are the weak topology on a Banach space X
and the weak* topology on its dual space X ∗ . With these topologies X
and X ∗ are locally convex Hausdorff topological vector spaces and the chapter
begins with a discussion of the elementary properties of such spaces. The
central result of the third chapter is the Banach–Alaoglu Theorem which CONTENTS 3 asserts that the unit ball in the dual space is compact with respect to the
weak* topology. This theorem has important consequences in many fields of
mathematics. The chapter also contains a proof of the Banach–Dieudonn´e
Theorem which asserts that a linear subspace of the dual space of a Banach
space is weak* closed if and only if its intersection with the closed unit
ball is weak* closed. A consequence of the Banach–Alaoglu Theorem is
that the unit ball in a reflexive Banach space is weakly compact, and the
ˇ
Eberlein–Smulyan
Theorem asserts that this property characterizes reflexive
Banach spaces. The Kreˇın–Milman Theorem asserts that every nonempty
compact convex subset of a locally convex Hausdorff topological vector space
is the closed convex hull of its extremal points. Combining this with the
Banach–Alaoglu Theorem, one can prove that every homeomorphism of a
compact metric space admits an invariant ergodic Borel probability measure.
Some properties of such ergodic measures can be derived from an abstract
functional analytic ergodic theorem which is also established in this chapter.
The purpose of Chapter 4 is to give a basic introduction to Fredholm
operators and their indices including the stability theorem. A Fredholm
operator is a bounded linear operator between Banach spaces that has a
finite-dimensional kernel, a closed image, and a finite-dimensional cokernel.
Its Fredholm index is the difference of the dimensions of kernel and cokernel.
The stability theorem asserts that the Fredholm operators of any given index
form an open subset of the space of all bounded linear operators between two
Banach spaces, with respect to the topology induced by the operator norm.
It also asserts that the sum of a Fredholm operator and a compact operator is
again Fredholm and has the same index as the original operator. The chapter
includes an introduction to the dual of a bounded linear operator, a proof of
the closed image theorem which asserts that an operator has a closed image
if and only if its dual does, an introduction to compact operators which map
the unit ball to pre-compact subsets of the target space, a characterization
of Fredholm operators in terms of invertibility modulo compact operators,
and a proof of the stability theorem for Fredholm operators.
The purpose of Chapter 5 is to study the spectrum of a bounded linear
operator on a real or complex Banach space. A first preparatory section
discusses complex Banach spaces and the complexifications of real Banach
spaces, the integrals of continuous Banach space valued functions on compact intervals, and holomorphic operator valued functions. The chapter then
introduces the spectrum of a bounded linear operator, examines its elementary properties, discusses the spectra of compact operators, and establishes 4 CONTENTS the holomorphic functional calculus. The remainder of this chapter deals
exclusively with operators on Hilbert spaces, starting with a discussion of
complex Hilbert spaces and the spectra of normal and self-adjoint operators. It then moves on to C ∗ algebras and the continuous functional calculus
for self-adjoint operators, which takes the form of an isomorphism from the
C* algebra of complex valued continuous functions on the spectrum to the
smallest C* algebra containing the given operator. The next topic is the
Gelfand representation and the extension of the continuous functional calculus to normal operators. The chapter also contains a proof that every
normal operator can be represented by a projection valued measure on the
spectrum, and that every self-adjoint operator is isomorphic to a direct sum
of multiplication operators on L2 spaces.
Chapter 6 is devoted to unbounded operators and their spectral theory.
The domain of an unbounded operator on a Banach space is a linear subspace.
In most of the relevant examples the domain is dense and the operator has a
closed graph. The chapter includes a discussion of the dual of an unbounded
operator and an extension of the closed image theorem to this setting. It
then examines the basic properties of the spectra of unbounded operators.
The remainder of the chapter deals with unbounded operators on Hilbert
spaces and their adjoints. In particular, it extends the functional calculus
and the spectral measure to unbounded self-adjoint operators.
Stongly continuous semigroups of operators are the subject of Chapter 7.
They play an important role in the study of many linear partial differential
equations such as the heat equation, the wave equation, and the Schr¨odinger
equation, and they can be viewed as infinite-dimensional analogues of the
exponential matrix S(t) := etA . In all the relevant examples the operator A
is unbounded. It is called the infinitesimal generator of the strongly continuous semigroup in question. A central result in the subject is the Hille–
Yosida–Phillips Theorem which characterizes the infinitesimal generators of
strongly continuous semigroups. The dual semigroup is not always strongly
continuous. It is, however, strongly continuous whenever the Banach space
in question is reflexive. The proof requires an understanding of the subtle
properties of strongly and weakly measurable functions with values in a Banach space. The chapter closes with a study of analytic semigroups and their
infinitesimal generators.
Each of the s...
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