دانلود کتاب Lectures on Functional Analysis and the Lebesgue Integral

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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