Book:Avner Friedman/Foundations of Modern Analysis
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Avner Friedman: Foundations of Modern Analysis
Published $\text {1970}$, Dover Publications
- ISBN 0-486-64062-0
Subject Matter
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
- 2.12 The Radon-Nikodym 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
- 4.13 Adjoint Operators
- 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
- 6.5 Self-Adjoint Operators
- 6.6 Positive Operators
- 6.7 Spectral Families of Self-Adjoint Operators
- 6.8 The Resolvent of Self-Adjoint Operators
- 6.9 Eigenvalue Problems for Differential Equations
- Bibliogrpahy
- Index
Source work progress
- 1970: Avner Friedman: Foundations of Modern Analysis ... (previous) ... (next): $\S 1.1$: Rings and Algebras: Problem $1.1.3$