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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 6 9 17 22 22 24 29 31 31 32 38 42 47 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 58 62 62 66 70 73 73 76 v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . ˇ 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 82 86 87 88 88 89 93 94 109 . . . . . . . . . . . . . . . . . 117 . 118 . 118 . 123 . 127 . 130 . 132 . 132 . 133 . 134 . 138 . 142 . 148 . 152 . 152 . 153 . 155 . 161 . . . . . . . 171 . 172 . 172 . 174 . 178 . 183 . 189 . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 . 210 . 210 . 213 . 216 . 219 . 219 . 222 . 224 . 227 . 233 . 233 . 237 . 239 . 243 . 246 . 246 . 248 . 252 . 258 . 258 . 267 . 269 . 273 . 274 . 278 . 291 . 298 . . . . . . 305 . 305 . 305 . 309 . 315 . 316 . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 324 331 336 342 353 . . . . . . . . . . . . . . . . . . 359 . 360 . 360 . 363 . 366 . 373 . 373 . 378 . 384 . 387 . 387 . 392 . 394 . 398 . 403 . 403 . 408 . 417 . 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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