Book:Avner Friedman/Foundations of Modern Analysis

Subject Matter

 * Real Analysis

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
* : $\S 1.1$: Rings and Algebras: Problem $1.1.3$