### Transcription of FUNCTIONAL ANALYSIS - ETH Z

1 **FUNCTIONAL** ANALYSISTheo B uhlerETH Z urichDietmar A. SalamonETH Z urich8 June 2017iiPrefaceThese are notes for the lecture course **FUNCTIONAL** **ANALYSIS** I held by thesecond author at ETH Z urich in the fall semester 2015. Prerequisites arethe first year courses onAnalysisandLinear Algebra, and the second yearcourses onComplex **ANALYSIS** ,Topology, andMeasure and material of Subsection on elementary Hilbert space theory, Sub-section on the Stone Weierstra Theorem, and the appendices on theLemma of Zorn and Tychonoff s Theorem has not been covered in the lec-tures. These topics were assumed to have been covered in previous lecturecourses. They are included here for completeness of the material of Subsection on the James space, Section on thefunctional calculus for bounded normal operators, Chapter 6 on unboundedlinear operators, Subsection on Banach space valuedLpfunctions, Sub-section on self-adjoint and unitary semigroups, and Section on an-alytic semigroups was not part of the lecture course (with the exception ofsome of the basic definitions in Chapter 6 that are relevant for infinitesimalgenerators of strongly continuous semigroups, namely, parts of Section onthe dual of an unbounded operator on a Banach space and Subsection the adjoint of an unbounded operator on a Hilbert space).

2 7 June 2017 Theo B uhlerDietmar A. SalamoniiiivContentsIntroduction11 Metric Spaces and Compact Sets .. Spaces .. Sets .. Arzel`a Ascoli Theorem .. Finite-Dimensional Banach Spaces .. Linear Operators .. Normed Vector Spaces .. and Product Spaces .. The Dual Space .. Banach Space of Bounded Linear Operators .. of Dual Spaces .. Spaces .. Banach Algebras .. The Baire Category Theorem .. Problems .. 522 Principles of **FUNCTIONAL** Uniform Boundedness .. Open Mappings and Closed Graphs .. Open Mapping Theorem .. Closed Graph Theorem .. Operators .. Hahn Banach and Convexity .. Hahn Banach Theorem .. Linear Functionals .. of Convex Sets .. Closure of a Linear Subspace .. Subspaces .. Bases .. Reflexive Banach Spaces .. Bidual Space.

3 Banach Spaces .. Banach Spaces .. James Space .. Problems .. 1093 The Weak and Weak* Topological Vector Spaces .. and Examples .. Sets .. Properties of the Weak Topology .. Properties of the Weak* Topology .. The Banach Alaoglu Theorem .. Separable Case .. Measures .. General Case .. The Banach Dieudonn e Theorem .. The Eberlein Smulyan Theorem .. The Kre n Milman Theorem .. Ergodic Theory .. Measures .. and Times Averages .. Abstract Ergodic Theorem .. Problems .. 1614 Fredholm The Dual Operator .. and Examples .. Closed Image Theorem .. Compact Operators .. Fredholm Operators .. Composition and Stability .. Problems .. 2005 Spectral Complex Banach Spaces .. and Examples .. Functions .. The Spectrum .. Spectrum of a Bounded Linear Operator.

4 Spectral Radius .. Spectrum of a Compact Operator .. **FUNCTIONAL** Calculus .. Operators on Hilbert Spaces .. Hilbert Spaces .. Adjoint Operator .. Spectrum of a Normal Operator .. Spectrum of a Self-Adjoint Operator .. The Spectral Mapping Theorem .. * Algebras .. Stone Weierstra Theorem .. Calculus for Self-Adjoint Operators .. Spectral Representations .. Gelfand Representation .. * Algebras of Normal Operators .. Calculus for Normal Operators .. Spectral Measures .. Valued Measures .. **FUNCTIONAL** Calculus .. Cyclic Vectors .. Problems .. 2986 Unbounded Unbounded Operators on Banach Spaces .. and Examples .. Spectrum of an Unbounded Operator .. Projections .. The Dual of an Unbounded Operator .. Unbounded Operators on Hilbert Spaces .. Adjoint of an Unbounded Operator .. Self-Adjoint Operators.

5 Normal Operators .. **FUNCTIONAL** Calculus .. Spectral Measures .. Problems .. 3537 Semigroups of Strongly Continuous Semigroups .. and Examples .. Properties .. Infinitesimal Generator .. The Hille Yosida Phillips Theorem .. Cauchy Problems .. Hille Yosida Phillips Theorem .. Semigroups .. Semigroups and Duality .. Space Valued Measurable Functions .. Banach SpaceLp(I,X) .. Dual Semigroup .. on Hilbert Spaces .. Analytic Semigroups .. of Analytic Semigroups .. of Analytic Semigroups .. of Analytic Semigroups .. Problems .. 419A The Lemma of Zorn421B Tychonoff s Theorem427 References431 Notation435 Index437 IntroductionClassically, **FUNCTIONAL** **ANALYSIS** is the study of function spaces and linear op-erators between them. The relevant function spaces are often equipped withthe structure of a Banach space and many of the central results remain validin the more general setting of bounded linear operators between Banachspaces or normed vector spaces, where the specific properties of the concretefunction space in question only play a minor role.

6 Thus, in the modern guise, **FUNCTIONAL** **ANALYSIS** is the study of Banach spaces and bounded linear opera-tors between them, and this is the viewpoint taken in the present area of mathematics has both an intrinsic beauty, which we hope toconvey to the reader, and a vast number of applications in many fields ofmathematics. These include the **ANALYSIS** of PDEs, differential topology andgeometry, symplectic topology, **quantum** **mechanics** , probability theory, geo-metric group theory, dynamical systems, ergodic theory, and approximationtheory, among many others. While we say little about specific applications,they do motivate the choice of topics covered in this book, and our goal isto give a self-contained exposition of the necessary background in abstractfunctional **ANALYSIS** for many of the relevant manuscript is addressed primarily to third year students of mathe-matics or physics, and the reader is assumed to be familiar with first yearanalysis and linear algebra, as well as complex **ANALYSIS** and the basics of pointset topology and measure and integration.

7 For example, this manuscript doesnot include a proof of completeness and duality are naturally many topics that go beyond the scope of the presentmanuscript, such as Sobolev spaces and PDEs, which would require a book onits 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 thediscussion to linear operators and say nothing about nonlinear functionalanalysis. Other topics not covered include the Fourier transform (see [2, 32,12 CONTENTS54]), maximal regularity for semigroups (see [51]), the space of Fredholmoperators on an infinite-dimensional Hilbert space as a classifying space forK-theory (see [5, 6, 7, 28]), Quillen s determinant line bundle over the space ofFredholm operators (see [46, 52]), and the work of Gowers [17] and Argyros Haydon [4] on Banach spaces on which every bounded linear operator is thesum of scalar multiple of the identity and a compact operator.

8 Here is adescription of the content of the book, chapter by 1 discusses some basic concepts that play a central role in thesubject. It begins with a section on metric spaces and compact sets whichincludes a proof of the Arzel`a Ascoli theorem. It then moves on to establishsome basic properties of finite-dimensional normed vector space spaces andshows, in particular, that a normed vector space is finite-dimensional if andonly if the unit ball is compact. The first chapter also introduces the dualspace of a normed vector space, explains several important examples, andcontains an introduction to elementary Hilbert space theory. It then intro-duces Banach algebras and shows that the group of invertible elements is anopen set. It closes with a proof of the Baire category 2 is devoted to the three fundamentalprinciples of functionalanalysis. They are theUniform Boundedness Principle(a pointwise boundedfamily of bounded linear operators on a Banach space is bounded), theOpenMapping Theorem(a surjective bounded linear operator between Banachspaces is open), and theHahn Banach Theorem(a bounded linear func-tional on a linear subspace of a normed vector space extends to a boundedlinear **FUNCTIONAL** on the entire normed vector space).

9 An equivalent formu-lation of the Open Mapping Theorem is theClosed Graph Theorem(a linearoperator between Banach spaces is bounded if and only if it has a closedgraph) and a corollary is theInverse Operator Theorem(a bijective boundedlinear operator between Banach spaces has a bounded inverse). A slightlystronger version of the Hahn Banach theorem, with the norm replaced bya quasi-seminorm, can be reformulated as the geometric assertion that twoconvex subsets of a normed vector space can be separated by a hyperplanewhenever one of them has nonempty interior. The chapter also discussesreflexive Banach spaces and includes an exposition of the James subject of Chapter 3 are the weak topology on a Banach spaceXand the weak* topology on its dual spaceX . With these topologiesXandX are locally convex Hausdorff topological vector spaces and the chapterbegins with a discussion of the elementary properties of such spaces.

10 Thecentral result of the third chapter is the Banach Alaoglu Theorem whichCONTENTS3asserts that the unit ball in the dual space is compact with respect to theweak* topology. This theorem has important consequences in many fields ofmathematics. The chapter also contains a proof of the Banach Dieudonn eTheorem which asserts that a linear subspace of the dual space of a Banachspace is weak* closed if and only if its intersection with the closed unitball is weak* closed. A consequence of the Banach Alaoglu Theorem isthat the unit ball in a reflexive Banach space is weakly compact, and theEberlein Smulyan Theorem asserts that this property characterizes reflexiveBanach spaces. The Kre n Milman Theorem asserts that every nonemptycompact convex subset of a locally convex Hausdorff topological vector spaceis the closed convex hull of its extremal points.