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

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## Serge Lang:

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

Published $\text {1993}$, **Springer**

- ISBN 978-1461269380.

### Subject Matter

### 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**