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

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