FUNCTIONAL ANALYSIS - People

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.

2 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).

3 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.

4 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 .. 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.

5 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 .. Spectral Radius .. Spectrum of a Compact operator .. FUNCTIONAL Calculus .. Operators on Hilbert Spaces .. Hilbert Spaces .. Adjoint operator .

6 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.

7 Adjoint of an Unbounded operator .. Self-Adjoint Operators .. 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.

8 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. 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.

9 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.

10 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 .