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

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

Published $\text {1993}$, Springer

ISBN 978-1461269380.

### 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
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