Lectures on Functional Analysis and the Lebesgue Integral

Description

This textbook, based on three series of lectures held by the author at the University of Strasbourg, presents functional analysis in a non-traditional way by generalizing elementary theorems of plane geometry to spaces of arbitrary dimension. This approach leads naturally to the basic notions and theorems. Most results are illustrated by the small lp spaces. The Lebesgue integral, meanwhile, is treated via the direct approach of Frigyes Riesz, whose constructive definition of measurable functions leads to optimal, clear-cut versions of the classical theorems of Fubini-Tonelli and Radon-Nikodým. Lectures on Functional Analysis and the Lebesgue Integral presents the most important topics for students, with short, elegant proofs. The exposition style follows the Hungarian mathematical tradition of Paul Erdos and others. The order of the first two parts, functional analysis and the Lebesgue integral, may be reversed. In the third and final part they are combined to study various spaces of continuous and integrable functions. Several beautiful, but almost forgotten, classical theorems are also included. Both undergraduate and graduate students in pure and applied mathematics, physics and engineering will find this textbook useful. Only basic topological notions and results are used and various simple but pertinent examples and exercises illustrate the usefulness and optimality of most theorems. Many of these examples are new or difficult to localize in the literature, and the original sources of most notions and results are indicated to help the reader understand the genesis and development of the field. Preface Contents Topological Prerequisites Topological Spaces Metric Spaces Normed Spaces Part I Functional Analysis 1 Hilbert Spaces 1.1 Definitions and Examples 1.2 Orthogonality 1.3 Separation of Convex Sets: Theorems of Riesz–Fréchet and Kuhn–Tucker 1.4 Orthonormal Bases 1.5 Weak Convergence: Theorem of Choice 1.6 Continuous and Compact Operators 1.7 Hilbert's Spectral Theorem 1.8 * The Complex Case 1.9 Exercises 2 Banach Spaces 2.1 Separation of Convex Sets 2.2 Theorems of Helly–Hahn–Banach and Taylor–Foguel 2.3 The p Spaces and Their Duals 2.4 Banach Spaces 2.5 Weak Convergence: Helly–Banach–Steinhaus Theorem 2.6 Reflexive Spaces: Theorem of Choice 2.7 Reflexive Spaces: Geometrical Applications 2.8 * Open Mappings and Closed Graphs 2.9 * Continuous and Compact Operators 2.10 * Fredholm–Riesz Theory 2.11 * The Complex Case 2.12 Exercises 3 Locally Convex Spaces 3.1 Families of Seminorms 3.2 Separation and Extension Theorems 3.3 Krein–Milman Theorem 3.4 * Weak Topology. Farkas–Minkowski Lemma 3.5 * Weak Star Topology: Theorems of Banach–Alaoglu and Goldstein 3.6 * Reflexive Spaces: Theorems of Kakutaniand Eberlein–Šmulian 3.7 * Topological Vector Spaces 3.8 Exercises Part II The Lebesgue Integral 4 * Monotone Functions 4.1 Continuity: Countable Sets 4.2 Differentiability: Null Sets 4.3 Jump Functions 4.4 Proof of Lebesgue's Theorem 4.5 Functions of Bounded Variation 4.6 Exercises 5 The Lebesgue Integral in R 5.1 Step Functions 5.2 Integrable Functions 5.3 The Beppo Levi Theorem 5.4 Theorems of Lebesgue, Fatou and Riesz–Fischer 5.5 * Measurable Functions and Sets 5.6 Exercises 6 * Generalized Newton–Leibniz Formula 6.1 Absolute Continuity 6.2 Primitive Function 6.3 Integration by Parts and Change of Variable 6.4 Exercises 7 Integrals on Measure Spaces 7.1 Measures 7.2 Integrals Associated with a Finite Measure 7.3 Product Spaces: Theorems of Fubini and Tonelli 7.4 Signed Measures: Hahn and Jordan Decompositions 7.5 Lebesgue Decomposition 7.6 The Radon–Nikodým Theorem 7.7 * Local Measurability 7.8 Exercises Part III Function Spaces 8 Spaces of Continuous Functions 8.1 Weierstrass Approximation Theorems 8.2 * The Stone–Weierstrass Theorem 8.3 Compact Sets. The Arzelà–Ascoli Theorem 8.4 Divergence of Fourier Series 8.5 Summability of Fourier Series. Fejér's Theorem 8.6 * Korovkin's Theorems. Bernstein Polynomials 8.7 * Theorems of Haršiladze–Lozinski, Nikolaev and Faber 8.8 * Dual Space. Riesz Representation Theorem 8.9 Weak Convergence 8.10 Exercises 9 Spaces of Integrable Functions 9.1 Lp Spaces, 1≤p≤∞ 9.2 * Compact Sets 9.3 * Convolution 9.4 Uniformly Convex Spaces 9.5 Reflexivity 9.6 Duals of Lp Spaces 9.7 Weak and Weak Star Convergence 9.8 Exercises 10 Almost Everywhere Convergence 10.1 Lp Spaces, 1≤p≤∞ 10.2 Lp Spaces, 0< p≤1 10.3 L0 Spaces 10.4 Convergence in Measure Hints and Solutions to Some Exercises Teaching Remarks Functional Analysis The Lebesgue Integral Function Spaces Bibliography Subject Index Name Index