Book:Serge Lang/Real and Functional Analysis/Third Edition

Serge Lang: Real and Functional Analysis (3rd Edition)

Published $\text {1993}$, Springer

ISBN 978-1461269380

Subject Matter

• Analysis

• Real Analysis

• Functional Analysis

Contents

Part One: General Topology

Chapter I: Sets

Some Basic Terminology

Denumberable Sets

Zorn's Lemma

Chapter II: Topological Spaces

Open and Closed Sets

Connected Sets

Compact Spaces

Separation by Continuous Functions

Exercises

Chapter III: Continuous Functions on Compact sets

The Stone-Weierstrass Theorem

Ideals of Continuous Functions

Ascoli's Theorem

Exercises

Part Two:Banach and Hilbert Spaces

Chapter IV: Banach Spaces

Definitions, the Dual Space, and the Hahn-Banach Theorem

Banach Algebras

The Linear Extension Theorem

Completion of a Normed Theorem

Completion of a Normed Vector Space

Spaces with Operators

Appendix: Convex Sets

The Krein-Milman Theorem

Mazur's Theorem

Exercises

Chapter V: Hilbert Space

Hermitian Forms

Functional and Operators

Exercises

Part Three: Integration

Chapter VI: The General Integral

Measured Spaces, Measurable Maps, and Positive Measures

The Integral of Step Maps

The $L^1$-Completion

Properties of the Integral: First Part

Properties of the Integral: Second Part

Approximations

Extension of Positive Measures from Algebras to $\sigma$-Algebras

Product Measures and Integration on a Product Space

The Lebesgue Integral in $\R^p$

Exercises

Chapter VII: Duality and Representation Theorems

The Hilbert Space $\map {L^2} \mu$

Duality Between $\map {L^1} \mu$ and $\map {L^\infty} \mu$

Complex and Vectorial Measures

Complex or Vectorial Measures and Duality

The $L^p$ Spaces, $1 < p < \infty$

The Law of Large Numbers

Exercises

Chapter VIII: Some Applications of Integration

Convolution

Continuity and Differentiation Under the Integral Sign

Dirac Sequences

The Schwarz Space and Fourier Transform

The Fourier Inversion Formula

The Poisson Summation Formula

An Example of Fourier Transform Not in the Schwartz Space

Exercises

Chapter IX: Integration and Measures on Locally Compact Spaces

Positive and Bounded Functionals on $\map {C_c} X$

Positive Functional as Integrals

Regular Positive Measures

Bounded Functionals as Integrals

Localization of a Measure and of the Integral

Product Measures on Locally Compact Spaces

Exercises

Chapter X: Riemann-Stieltjes Integral and Measure

Functions of Bounded Variation and the Stieltjes Integral

Applications to Fourier Analysis

Exercises

Chapter XI: Distributions

Definition and Examples

Support and Examples

Derivation of Distributions

Distributions with Discrete Support

Chapter XII: Integration on Locally Compact Groups

Topological Groups

The Haar Integral, Uniqueness

Existence of the Haar Integral

Measures on Factor Groups and Homogeneous Spaces

Exercises

Part Four: Calculus

Chapter XIII: Differential Calculus

Integration in One Variable

The Derivative as a Linear Map

Properties of the Derivative

Mean Value Theorem

The Second Derivative

Higher Derivatives and Taylor's Formula

Partial Derivatives

Differentiation Under the Integral Sign

Differentiation of Sequences

Exercises

Chapter XIV: Inverse Mapping and Differential Equations

The Inverse Mapping Theorem

The Implicit Mapping Theorem

Existence Theorem for Differential Equations

Local Dependence on Initial Conditions

Global Smoothness of the Flow

Exercises

Part Five: Functional Analysis

Chapter XV: The Open Mapping Theorem, Factor Spaces, and Duality

The Open Mapping Theorem

Orthogonality

Applications of the Open Mapping Theorem

Chapter XVI: The Spectrum

The Gelfan-Mazur Theorem

The Gelfand Transform

$C^*$-Algebras

Exercises

Chapter XVII: Compact and Fredholm Operators

Compact Operators

Fredholm Operators and the Index

Spectral Theorem for Compact Operators

Application to Integral Equations

Exercises

Chapter XVIII: Spectral Theorem for Bounded Hermitian Operators

Hermitian and Unitary Operators

Positive Hermitian Operators

The Spectral Theorem for Compact Hermitian Operators

The Spectral Theorem for Hermitian Operators

Orthogonal Projections

Schur's Lemma

Polar Decomposition of Endomorphisms

The Morse-Palais Lemma

Exercises

Chapter XIX: Further Spectral Theorems

Projection Functions of Operators

Self-Adjoint Operators

Example: The Laplace Operator in the Plane

Chapter XX: Spectral Measures

Definition of the Spectral Measure

Uniqueness of the Spectral Measure: the Titchmarsh-Kodaira Formula

Unbounded Functions of Operators

Spectral Families of Projections

The Spectral Integral as Stieltjes Integral

Exercises

Part Six: Global Analysis

Chapter XXI: Local Integration of Differential Forms

Sets of Measure 0

Change of Variables Formula

Differential Forms

Inverse Image of a Form

Appendix

Chapter XXII: Manifolds

Atlases, Charts, Morphisms

Submanifolds

Tangent Spaces

Partitions of Unity

Manifolds with Boundary

Vector Fields and Global Differential Equations

Chapter XXIII: Integration and Measures on Manifolds

Differential Forms on Manifolds

Orientation

The Measure Associated with a Differential Form

Stokes' Theorem for a Rectangular Simplex

Stokes' Theorem on a Manifold

Stokes' Theorem with Singularities

Bibliography

Table of Notation

Index