FUNCTIONAL ANALYSIS - ETH Z

1 FUNCTIONAL ANALYSISTheo B uhlerDietmar A. SalamonETH Z urichE-mail Subject 46-01 47-01 Key words and space, bounded linear operator , dualspace, dual operator , Arzel`a Ascoli, uniform boundedness, open mapping,closed graph, Hahn Banach, James space, weak and weak* topology,Banach Alaoglu, Banach Dieudonn e, Eberlein- Smulyan, Kre n Milman,ergodic theorem, closed image theorem, compact operator , Fredholmtheory, spectral theory, FUNCTIONAL calculus, Gelfand representation,spectral measure, unbounded operator , strongly continuous semigroup,infinitesimal generator, Hille Yosida Phillips, analytic book provides an introduction to the subject of Func-tional ANALYSIS for third year students of mathematics and physics witha basic knowledge of first year ANALYSIS and linear algebra as well as somecomplex ANALYSIS , point set topology, and measure and 1.

2 Foundations1 Metric Spaces and Compact Sets2 Finite-Dimensional Banach Spaces17 The Dual Space25 Hilbert Spaces31 Banach Algebras35 The Baire Category Theorem40 Problems45 Chapter 2. Principles of FUNCTIONAL Analysis49 Uniform Boundedness50 Open Mappings and Closed Graphs54 Hahn Banach and Convexity65 Reflexive Banach Spaces80 Problems101 Chapter 3. The Weak and Weak* Topologies109 Topological Vector Spaces110 The Banach Alaoglu Theorem124 The Banach Dieudonn e Theorem130 The Eberlein Smulyan Theorem134vviContents The Kre n Milman Theorem140 Ergodic Theory144 Problems153 Chapter 4. Fredholm Theory163 The Dual Operator164 Compact Operators173 Fredholm Operators179 Composition and Stability184 Problems189 Chapter 5. Spectral Theory197 Complex Banach Spaces198 Spectrum208 Operators on Hilbert Spaces222 FUNCTIONAL Calculus for Self-Adjoint Operators234 Gelfand Spectrum and Normal Operators246 Spectral Measures261 Cyclic Vectors281 Problems288 Chapter 6.

3 Unbounded Operators295 Unbounded Operators on Banach Spaces295 The Dual of an Unbounded Operator306 Unbounded Operators on Hilbert Spaces313 FUNCTIONAL Calculus and Spectral Measures326 Problems342 Chapter 7. Semigroups of Operators349 Strongly Continuous Semigroups350 The Hille Yosida Phillips Theorem363 The Dual Semigroup377 Analytic Semigroups388 Banach Space Valued Measurable Functions404 Inhomogeneous Equations425 Problems439 Appendix A. Zorn and Tychonoff445 Contentsvii The Lemma of Zorn445 Tychonoff s Theorem450 Bibliography453 Notation457 Index461 PrefaceThese 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 Section on elementary Hilbert space theory, Sub-section on the Stone Weierstra Theorem, and the appendix on theLemma of Zorn and Tychonoff s Theorem was not covered in the topics were assumed to have been covered in previous lecture are included here for completeness of the material of Subsection on the James space, Section on thefunctional calculus for bounded normal operators, and Chapter 6 on un- bounded operators 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).

4 From Chapter 7 only thebasic material on strongly continuous semigroups in Section , on theirinfinitesimal generators in Section , and on the dual semigroup in Sec-tion were included in the lecture February 2018 Theo B uhlerDietmar A. SalamonixIntroductionClassically, FUNCTIONAL ANALYSIS is the study of function spaces and linearoperators between them. The relevant function spaces are often equippedwith the structure of a Banach space and many of the central results re-main valid in the more general setting of bounded linear operators betweenBanach spaces or normed vector spaces, where the specific properties ofthe concrete function space in question only play a minor role. Thus, in themodern guise, FUNCTIONAL ANALYSIS is the study of Banach spaces and boundedlinear operators between them, and this is the viewpoint taken in the presentbook.

5 This area of mathematics has both an intrinsic beauty, which we hopeto convey 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 book is addressed primarily to third year students of mathematicsor physics, and the reader is assumed to be familiar with first year analysisand linear algebra, as well as complex ANALYSIS and the basics of point settopology and measure and integration.

6 For example, this book does notinclude a proof of completeness and duality are naturally many topics that go beyond the scope of the presentbook, such as Sobolev spaces and PDEs, which would require a book onits own and, in fact, very many books have been written about this sub-ject; here we just refer the interested reader to [19, 28, 30]. We alsoxixiiIntroductionrestrict the discussion to linear operators and say nothing about nonlinearfunctional ANALYSIS . Other topics not covered include the Fourier transform(see [2, 48, 79]), maximal regularity for semigroups (see [76]), the spaceof Fredholm operators on an infinite-dimensional Hilbert space as a clas-sifying space for K-theory (see [5, 6, 7, 42]), Quillen s determinant linebundle over the space of Fredholm operators (see [71, 77]), and the workof Gowers [31] and Argyros Haydon [4] on Banach spaces on which everybounded linear operator is the sum of a scalar multiple of the identity and acompact operator .

7 Here is a description of the contents of the book, chapterby 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 spaces and shows,in particular, that a normed vector space is finite-dimensional if and only ifthe unit ball is compact. The first chapter also introduces the dual space of anormed vector space, explains several important examples, and contains anintroduction to elementary Hilbert space theory. It then introduces Banachalgebras and shows that the group of invertible elements is an open set. Itcloses 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).

8 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 subjects 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 chap-ter begins with a discussion of the elementary properties of such spaces.

9 Thecentral result of the third chapter is the Banach Alaoglu Theorem whichIntroductionxiiiasserts 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 unit ballis weak* closed. A consequence of the Banach Alaoglu Theorem is that theunit ball in a reflexive Banach space is weakly compact, and the Eberlein Smulyan Theorem asserts that this property characterizes reflexive Banachspaces. The Kre n Milman Theorem asserts that every nonempty compactconvex subset of a locally convex Hausdorff topological vector space is theclosed convex hull of its extremal points.

10 Combining this with the Banach Alaoglu Theorem, one can prove that every homeomorphism of a compactmetric space admits an invariant ergodic Borel probability measure. Someproperties of such ergodic measures can be derived from an abstract func-tional analytic ergodic theorem which is also established in this purpose of Chapter 4 is to give a basic introduction to Fredholmoperators and their indices including the stability theorem. A Fredholmoperator is a bounded linear operator between Banach spaces that has afinite-dimensional kernel, a closed image, and a finite-dimensional Fredholm index is the difference of the dimensions of kernel and stability theorem asserts that the Fredholm operators of any given indexform an open subset of the space of all bounded linear operators between twoBanach spaces, with respect to the topology induced by the operator also asserts that the sum of a Fredholm operator and a compact operator isagain Fredholm and has the same index as the original operator .