Topics in Real and Functional Analysis
Gerald Teschl
Abstract
This manuscript provides a brief introduction to Real and (linear and nonlinear) Functional Analysis.
It covers basic Hilbert and Banach space theory as well as basic measure theory including
Lebesgue spaces and the Fourier transform.
MSC: 4601, 2801, 46E30
Keywords: Functional Analysis, Banach space, Hilbert space, Measure theory, Lebesgue spaces, Fourier transform, Mapping degree, fixedpoint theorems,
differential equations, NavierStokes equation
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Table of contents

Preface
 Warm up: Metric and topological spaces
 Basics
 Convergence and completeness
 Functions
 Product topologies
 Compactness
 Connectedness
 Separation
 Continuous functions on metric spaces
 A first look at Banach and Hilbert spaces
 Introduction: Linear partial differential equations
 The Banach space of continuous functions
 The geometry of Hilbert spaces
 Completeness
 Bounded operators
 Sums and quotients of Banach spaces
 Hilbert spaces
 Hilbert spaces
 The projection theorem and the Riesz lemma
 Operators defined via forms
 Orthogonal sums and tensor products
 Compact operators
 Compact operators
 The spectral theorem for compact symmetric operators
 Applications to SturmLiouville operators
 Estimating eigenvalues
 The main theorems about Banach spaces
 The Baire theorem and its consequences
 The HahnBanach theorem and its consequences
 The adjoint operator
 The geometric HahnBanach theorem
 Weak convergence
 Weak topologies
 Beyond Banach spaces: Locally convex spaces
 More on Compact operators
 Canonical form of compact operators
 HilbertSchmidt and trace class operators
 Fredholm theory for compact operators
 Bounded linear operators
 Banach algebras
 The C^{*} algebra of operators and the spectral theorem
 Spectral measures
 The StoneWeierstraß theorem
 The Gelfand representation theorem
 Measures
 The problem of measuring sets
 Sigma algebras and measures
 Extending a premeasure to a measure
 Borel measures
 Measurable functions
 How wild are measurable objects
 Appendix: Jordan measurable sets
 Appendix: Equivalent definitions for the outer Lebesgue measure
 Integration
 Integration  Sum me up, Henri
 Product measures
 Transformation of measures and integrals
 Appendix: Transformation of LebesgueStieltjes integrals
 Appendix: The connection with the Riemann integral
 The Lebesgue spaces L^{p}
 Functions almost everywhere
 Jensen ≤ Hölder ≤ Minkowski
 Nothing missing in L^{p}
 Approximation by nicer functions
 Integral operators
 More measure theory
 Decomposition of measures
 Derivatives of measures
 Complex measures
 Hausdorff measure
 Infinite product measures
 The Bochner integral
 Weak and vague convergence of measures
 Appendix: Functions of bounded variation and absolutely continuous functions
 The dual of L^{p}
 The dual of L^{p}, p<∞
 The dual of L^{∞} and the Riesz representation theorem
 The RieszMarkov representation theorem
 The Fourier transform
 The Fourier transform on L^{1} and L^{2}
 Applications to linear partial differential equations
 Sobolev spaces
 Applications to evolution equations
 Tempered distributions
 Interpolation
 Interpolation and the Fourier transform on L^{p}
 The Marcinkiewicz interpolation theorem
 Analysis in Banach spaces
 Differentiation and integration in Banach spaces
 Contraction principles
 Ordinary differential equations
 The Brouwer mapping degree
 Introduction
 Definition of the mapping degree and the determinant formula
 Extension of the determinant formula
 The Brouwer fixed point theorem
 Kakutani's fixed point theorem and applications to game theory
 Further properties and extensions
 The Jordan curve theorem
 The LeraySchauder mapping degree
 The mapping degree on finite dimensional Banach spaces
 Compact operators
 The LeraySchauder mapping degree
 The LeraySchauder principle and the Schauder fixed point theorem
 Applications to integral and differential equations
 The stationary NavierStokes equation
 Introduction and motivation
 An insert on Sobolev spaces
 Existence and uniqueness of solutions
 Monotone operators
 Monotone operators
 The nonlinear LaxMilgram theorem
 The main theorem of monotone operators
 Appendix: Some set theory
Glossary of notations
Index