# Book:Avner Friedman/Foundations of Modern Analysis

## Avner Friedman: Foundations of Modern Analysis

Published $\text {1970}$, Dover Publications

ISBN 0-486-64062-0

### Contents

Preface
Chapter 1. Measure Theory
1.1 Rings and Algebras
1.2 Definition of Measure
1.3 Outer Measure
1.4 Construction of Outer Measure
1.5 Completion of Measures
1.6 The Lebesgue and the Lebesgue-Stieltjes Measures
1.7 Metric Spaces
1.8 Metric Outer Measure
1.9 Construction of Metric Outer Measures
1.10 Signed Measures
Chapter 2. Integration
2.1 Definition of Measurable Fuctions
2.2 Operations on Measurable Functions
2.3 Egoroff's Theorem
2.4 Convergence in Measure
2.5 Integrals of Simple Functions
2.6 Definition of the Integral
2.7 Elementary Properties of Integrals
2.8 Sequences of Integral Functions
2.9 Lebesgue's Bounded Convergence Theorem
2.10 Applications of Lebesgue's Bounded Convergence Theorem
2.11 The Riemann Integral
2.13 The Lebesgue Decomposition
2.14 The Lebesgue Integral on the Real Line
2.15 Product of Measures
2.16 Fubini's Theorem
Chapter 3. Metric Spaces
3.1 Topological and Metric Spaces
3.2 $L^p$ Spaces
3.3 Completion of Metric Spaces; $H^{m, p}$ Spaces
3.4 Complete Metric Spaces
3.5 Compact Metric Spaces
3.6 Continuous Functions on Metric Spaces
3.7 The Stone-Weierstrass Theorem
3.8 A Fixed-Point Theorem and Applications
Chapter 4. Elements of Functional Analysis in Banach Spaces
4.1 Linear Normed Spaces
4.2 Subspaces and Bases
4.3 Finite-Dimensional Normed Linear Spaces
4.4 Linear Transformations
4.5 The Principle of Uniform Boundedness
4.6 The Open-Mapping Theorem and the Closed-Graph Theorem
4.7 Applications to Partial Differential Equations
4.8 The Hahn-Banach Theorem
4.9 Applications to the Dirichlet Problem
4.10 Conjugate Spaces and Reflexive Spaces
4.11 Tychonoff's Theorem
4.12 Weak Topology in Conjugate Spaces
4.14 The Conjugates of $L^p$ and $C \sqbrk {0, 1}$
Chapter 5. Completely Continuous Operators
5.1 Basic Properties
5.2 The Fredholm-Riesz-Schauder Theory
5.3 Elements of Spectral Theory
5.4 Applications to the Dirichlet Problem
Chapter 6. Hilbert Spaces and Spectral Theory
6.1 Hilbert Spaces
6.2 The Projection Theorem
6.3 Projection Operators
6.4 Orthonormal Sets