A Course in Functional Analysis and Measure Theory

Description

Written by an expert on the topic and experienced lecturer, this textbook provides an elegant, self-contained introduction to functional analysis, including several advanced topics and applications to harmonic analysis. Starting from basic topics before proceeding to more advanced material, the book covers measure and integration theory, classical Banach and Hilbert space theory, spectral theory for bounded operators, fixed point theory, Schauder bases, the Riesz-Thorin interpolation theorem for operators, as well as topics in duality and convexity theory. Aimed at advanced undergraduate and graduate students, this book is suitable for both introductory and more advanced courses in functional analysis. Including over 1500 exercises of varying difficulty and various motivational and historical remarks, the book can be used for self-study and alongside lecture courses. Preface to the English Translation Contents Introduction 1 Metric and Topological Spaces 1.1 Sets and Maps 1.2 Topological Spaces 1.2.1 Terminology 1.2.2 The Product of Two Topological Spaces 1.2.3 Compact Spaces 1.2.4 Semicontinuous Functions 1.3 Metric Spaces 1.3.1 The Axioms of Metric. Sequences and Topology 1.3.2 Distance of a Point to a Set. Continuity of Distance 1.3.3 Completeness 1.3.4 Uniform Continuity. The Extension Theorem 1.3.5 Pseudometric Spaces and the Associated Metric Spaces. The Completion of a Metric Space 1.3.6 Sets of First Category and Baire's Theorem 1.4 Compact Sets in Metric Spaces 1.4.1 Precompact Sets 1.4.2 Spaces of Continuous Maps and Functions. Arzelà's Theorem 1.4.3 Application: The Isoperimetric Problem 1.4.4 The Cantor Set 2 Measure Theory 2.1 Systems of Sets and Measures 2.1.1 Algebras of Sets 2.1.2 sigma-Algebras of Sets. Borel Sets 2.1.3 Products of sigma-Algebras 2.1.4 Measures: Finite and Countable Additivity 2.1.5 Measure Spaces. Completeness. Completion of a sigma-Algebra with Respect to a Measure 2.1.6 Operations on Measures. delta-Measure. Atoms, Purely Atomic and Non-atomic Measures 2.2 Extension of Measures 2.2.1 Extension of a Measure from a Semiring of Sets to the Algebra Generated by the Semiring 2.2.2 Outer Measure 2.2.3 Extension of a Measure from an Algebra to a sigma-Algebra 2.2.4 A Monotone Class Theorem for Sets 2.3 Measures on an Interval and on the Real Line 2.3.1 The Lebesgue Measure on the Interval 2.3.2 A Bit More Terminology. The Meaning of the Term ``Almost Everywhere'' 2.3.3 Lebesgue's Theorem on the Differentiability of Monotone Functions 2.3.4 The Difficult Problem of Measure Theory. Existence of Sets that are not Lebesgue Measurable 2.3.5 Distribution Functions and the General form of a Borel Measure on the Interval 2.3.6 The Cantor Staircase and a Measure Uniformly Distributed on the Cantor Set 2.3.7 sigma-Finite Measures and the Lebesgue Measure on the Real Line 3 Measurable Functions 3.1 Measurable Functions and Operations on Them 3.1.1 Measurability Criterion 3.1.2 Elementary Properties of Measurable Functions 3.1.3 The Characteristic Function of a Set 3.1.4 Simple Functions. Lebesgue Approximation of Measurable Functions by Simple Ones. Measurability on the Completion of a Measure Space 3.2 Main Types of Convergence 3.2.1 Almost Everywhere Convergence 3.2.2 Convergence in Measure. Examples 3.2.3 Theorems Connecting Convergence in Measure to Convergence Almost Everywhere 3.2.4 Egorov's Theorem 4 The Lebesgue Integral 4.1 Convergence Along a Directed Set. Partitions 4.1.1 Directed Sets 4.1.2 Limit Along a Directed Set. Cauchy's Criterion 4.1.3 Partitions 4.2 Integrable Functions 4.2.1 Integral Sums 4.2.2 Definition and Simplest Properties of the Lebesgue Integral 4.2.3 The Integral as a Set Function 4.3 Measurability and Integrability 4.3.1 Measurability of Integrable Functions 4.3.2 The Uniform Limit Theorem 4.3.3 An Integrability Condition for Measurable Functions 4.4 Passage to the Limit Under the Integral Sign 4.4.1 Fatou's Lemma 4.4.2 Lebesgue's Dominated Convergence Theorem 4.4.3 Levi's Theorems on Sequences and Series 4.4.4 A Monotone Class Theorem for Functions 4.5 Multiple Integrals 4.5.1 Products of Measure Spaces 4.5.2 Double Integrals and Fubini's Theorem 4.5.3 A Converse to Fubini's Theorem 4.6 The Lebesgue Integral on an Interval and on the Real Line 4.6.1 The Lebesgue Integral and the Improper Integral on an Interval 4.6.2 The Integral with Respect to a sigma-finite Measure 4.6.3 Convolution 5 Linear Spaces, Linear Functionals, and the Hahn–Banach Theorem 5.1 Linear Spaces 5.1.1 Main Definitions 5.1.2 Ordered Sets and Zorn's Lemma 5.1.3 Existence of Hamel Bases 5.1.4 Linear Operations on Subsets 5.2 Linear Operators 5.2.1 Injectivity and Surjectivity 5.2.2 Quotient Spaces 5.2.3 Injectivization of a Linear Operator 5.3 Convexity 5.3.1 Definitions and Properties 5.3.2 Convex Hull 5.3.3 Hypersubspaces and Hyperplanes 5.4 The Hahn–Banach Theorem on the Extension of Linear Functionals 5.4.1 Convex Functionals 5.4.2 The Minkowski Functional 5.4.3 The Hahn–Banach Theorem—Analytic Form 5.5 Some Applications of the Hahn–Banach Theorem 5.5.1 Invariant Means on a Commutative Semigroup 5.5.2 The ``easy'' Problem of Measure Theory 6 Normed Spaces 6.1 Normed Spaces, Subspaces, and Quotient Spaces 6.1.1 Norms. Examples 6.1.2 The Metric of a Normed Space and Convergence. Isometries 6.1.3 The Space L1 6.1.4 Subspaces and Quotient Spaces 6.2 Connection Between the Unit Ball and the Norm. Lp Spaces 6.2.1 Properties of Balls in a Normed Space 6.2.2 Definition of the Norm by Means of a Ball. The Spaces Lp 6.3 Banach Spaces and Absolutely Convergent Series 6.3.1 Series. A Completeness Criterion in Terms of Absolute Convergence 6.3.2 Completeness of the Space L1 6.3.3 Subspaces and Quotient Spaces of Banach Spaces 6.4 Spaces of Continuous Linear Operators 6.4.1 A Continuity Criterion for Linear Operators 6.4.2 The Norm of an Operator 6.4.3 Pointwise Convergence 6.4.4 Completeness of the Space of Operators. Dual Space 6.5 Extension of Operators 6.5.1 Extension by Continuity 6.5.2 Projectors; Extension from a Closed Subspace 7 Absolute Continuity of Measures and Functions. The Connection Between Derivative and Integral 7.1 Charges. The Hahn and Radon–Nikodým Theorems 7.1.1 The Boundedness of Charges Theorem 7.1.2 The Hahn Decomposition Theorem 7.1.3 Absolutely Continuous Measures and Charges 7.1.4 The Charge Induced by a Function 7.1.5 Strong Singularity 7.1.6 The Radon–Nikodým Theorem 7.2 Derivative and Integral on an Interval 7.2.1 The Integral of a Derivative 7.2.2 The Derivative of an Integral as a Function of the Upper Integration Limit 7.2.3 Functions of Bounded Variation and the General Form of a Borel Charge on the Interval 7.2.4 Absolutely Continuous Functions 7.2.5 Absolutely Continuous Functions and Absolutely Continuous Borel Charges 7.2.6 Recovering a Function From its Derivative 7.2.7 Exercises: Change of Variables in the Lebesgue Integral 8 The Integral on C(K) 8.1 Regular Borel Measures on a Compact Space 8.1.1 Inner Measure and Regularity 8.1.2 The Support of a Measure 8.2 Extension of Elementary Integrals 8.2.1 Elementary Integrals 8.2.2 The Upper Integral of Lower Semi-continuous Functions 8.2.3 The Upper Integral on ellinfty(K) 8.2.4 The Space L(K,mathcalI) 8.3 Regular Borel Measures and the Integral 8.3.1 mathcalI-measurable Sets. The Measure Generated by an Integral 8.3.2 The General Form of Elementary Integrals 8.3.3 Approximation of Measurable Functions by Continuous Functions. Luzin's Theorem 8.4 The General Form of Linear Functionals on C(K) 8.4.1 Regular Borel Charges 8.4.2 Formulation of the Riesz–Markov–Kakutani Theorem. Uniqueness Theorem. Examples 8.4.3 Positive and Negative Parts of a Functional F inC(K)* 8.4.4 The Norm of a Functional on C(K) 8.4.5 Complex Charges and Integrals 8.4.6 Regular Complex Charges and Functionals on the Complex Space C(K) 9 Continuous Linear Functionals 9.1 The Hahn–Banach Theorem in Normed Spaces 9.1.1 The Connection Between Real and Complex Functionals 9.1.2 The Hahn–Banach Extension Theorem 9.2 Applications 9.2.1 Supporting Functionals 9.2.2 The Annihilator of a Subspace 9.2.3 Complete Systems of Elements 9.3 Convex Sets and the Hahn–Banach Theorem in Geometric Form 9.3.1 Some Lemmas 9.3.2 The Separation Theorem for Convex Sets 9.3.3 Examples 9.4 Adjoint Operators 9.4.1 The Connection Between Properties of an Operator and Those of Its Adjoint 9.4.2 The Duality Between Subspaces and Quotient Spaces 10 Classical Theorems on Continuous Operators 10.1 Open Mappings 10.1.1 An Openness Criterion 10.1.2 Ball-Like Sets 10.1.3 The Banach Open Mapping Theorem 10.2 Invertibility of Operators and Isomorphisms 10.2.1 Isomorphisms. Equivalent Norms 10.2.2 The Banach Inverse Operator Theorem 10.2.3 Bounded Below Operators. Closedness of the Image Criterion 10.3 The Graph of an Operator 10.3.1 The Closed Graph Theorem 10.3.2 Complemented Subspaces 10.4 The Uniform Boundedness Principle and Applications 10.4.1 The Banach–Steinhaus Theorem on Pointwise Bounded Families of Operators 10.4.2 Pointwise Convergence of Operators 10.4.3 Two Theorems on Fourier Series on an Interval 10.5 The Concept of a Schauder Basis 10.5.1 Definition and Simplest Properties 10.5.2 Coordinate Functionals and Partial Sum Operators 10.5.3 Linear Functionals on a Space with a Basis 11 Elements of Spectral Theory of Operators. Compact Operators 11.1 Algebra of Operators 11.1.1 Banach Algebras: Axiomatics and Examples 11.1.2 Invertibility in Banach Algebras 11.1.3 The Spectrum 11.1.4 The Resolvent and Non-emptyness of the Spectrum 11.1.5 The Spectrum and Eigenvalues of an Operator 11.1.6 The Matrix of an Operator 11.2 Compact Sets in Banach Spaces 11.2.1 Precompactness: General Results 11.2.2 Finite-Rank Operators and the Approximation Property 11.2.3 Compactness Criteria for Sets in Specific Spaces 11.3 Compact (Completely Continuous) Operators 11.3.1 Definition and Examples 11.3.2 Properties of Compact Operators 11.3.3 Operators of the Form I-T with T a Compact Operator 11.3.4 The Structure of the Spectrum of a Compact Operator 12 Hilbert Spaces 12.1 The Norm Generated by a Scalar Product 12.1.1 Scalar Product 12.1.2 The Cauchy–Schwarz Inequality 12.1.3 The Concept of Hilbert Space 12.2 Hilbert Space Geometry 12.2.1 The Best Approximation Theorem 12.2.2 Orthogonal Complements and Orthogonal Projectors 12.2.3 The General Form of Linear Functionals on a Hilbert Space 12.3 Orthogonal Series 12.3.1 A Convergence Criterion for Orthogonal Series 12.3.2 Orthonormal Systems. Bessel's Inequality 12.3.3 Fourier Series, Orthonormal Bases, and the Parseval Identity 12.3.4 Gram–Schmidt Orthogonalization and the Existence of Orthonormal Bases 12.3.5 The Isomorphism Theorem 12.4 Self-adjoint Operators 12.4.1 Bilinear Forms on a Hilbert Space 12.4.2 The Adjoint of a Hilbert Space Operator 12.4.3 Self-adjoint Operators and Their Quadratic Forms 12.4.4 Operator Inequalities 12.4.5 The Spectrum of a Self-adjoint Operator 12.4.6 Compact Self-adjoint Operators 13 Functions of an Operator 13.1 Continuous Functions of an Operator 13.1.1 Polynomials in an Operator 13.1.2 Polynomials in a Self-adjoint Operator 13.1.3 Definition of a Continuous Function of a Self-adjoint Operator 13.1.4 Properties of Continuous Functions of a Self-adjoint Operator 13.1.5 Applications of Continuous Functions of an Operator 13.2 Unitary Operators and the Polar Representation 13.2.1 The Absolute Value of an Operator 13.2.2 Definition and Simplest Properties of Unitary Operators 13.2.3 Polar Decomposition 13.3 Borel Functions of an Operator 13.4 Functions of a Self-adjoint Operator and the Spectral Measure 13.4.1 The Integral with Respect to a Vector Measure 13.4.2 Semivariation and Existence of the Integral 13.4.3 The Spectral Measure and Spectral Projectors 13.4.4 Linear Equations 14 Operators in Lp 14.1 Linear Functionals on Lp 14.1.1 The Hölder Inequality 14.1.2 Connections Between the Spaces Lp for Different Values of p 14.1.3 Weighted Integration Functionals 14.1.4 The General Form of Linear Functionals on Lp 14.2 The Fourier Transform on the Real Line 14.2.1 δ-Sequences and the Dini Theorem 14.2.2 The Fourier Transform in L1 on the Real Line 14.2.3 Inversion Formulas 14.2.4 The Fourier Transform and Differentiation 14.2.5 The Fourier Transform in L2 on the Real Line 14.3 The Riesz–Thorin Interpolation Theorem and its Consequences 14.3.1 The Hadamard Three-Lines Theorem 14.3.2 The Riesz–Thorin Theorem 14.3.3 Applications to Fourier Series and the Fourier Transform 15 Fixed Point Theorems and Applications 15.1 Some Classical Theorems 15.1.1 Contractive Mappings 15.1.2 The Fixed Point Property. Brouwer's Theorem 15.1.3 Partitions of Unity and Approximation of Continuous Mappings by Finite-Dimensional Mappings 15.1.4 The Schauder's Principle 15.2 Applications to Differential Equations and Operator Theory 15.2.1 The Picard and Peano Theorems on the Existence of a Solution to the Cauchy Problem for Differential Equations 15.2.2 The Lomonosov Invariant Subspace Theorem 15.3 Common Fixed Points of a Family of Mappings 15.3.1 Kakutani's Theorem 15.3.2 Topological Groups 15.3.3 Haar Measure 16 Topological Vector Spaces 16.1 Supplementary Material from Topology 16.1.1 Filters and Filter Bases 16.1.2 Limits, Limit Points, and Comparison of Filters 16.1.3 Ultrafilters. Compactness Criteria 16.1.4 The Topology Generated by a Family of Mappings. The Tikhonov Product 16.2 Background Material on Topological Vector Spaces 16.2.1 Axiomatics and Terminology 16.2.2 Completeness, Precompactness, Compactness 16.2.3 Linear Operators and Functionals 16.3 Locally Convex Spaces 16.3.1 Seminorms and Topology 16.3.2 Weak Topologies 16.3.3 Eidelheit's Interpolation Theorem 16.3.4 Precompactness and Boundedness 17 Elements of Duality Theory 17.1 Duality in Locally Convex Spaces 17.1.1 The General Notion of Duality. Polars 17.1.2 The Bipolar Theorem 17.1.3 The Adjoint Operator 17.1.4 Alaoglu's Theorem 17.2 Duality in Banach Spaces 17.2.1 w*-Convergence 17.2.2 The Second Dual 17.2.3 Weak Convergence in Banach Spaces 17.2.4 Total and Norming Sets. Metrizability Conditions 17.2.5 The Eberlein–Smulian Theorem 17.2.6 Reflexive Spaces 18 The Krein–Milman Theorem and Its Applications 18.1 Extreme Points of Convex Sets 18.1.1 Definitions and Examples 18.1.2 The Krein–Milman Theorem 18.1.3 Weak Integrals and the Krein–Milman Theorem in Integral Form 18.2 Applications 18.2.1 The Connection Between the Properties of the Compact Space K and Those of the Space C(K) 18.2.2 The Stone–Weierstrass Theorem 18.2.3 Completely Monotone Functions 18.2.4 Lyapunov's Theorem on Vector Measures References Index