- #1
- 22,097
- 3,277
Main Question or Discussion Point
- Author: Serge Lang
- Title: Real and Functional Analysis by Lang
- Amazon Link: https://www.amazon.com/dp/0387940014/?tag=pfamazon01-20
- Prerequisities: Undergrad analysis
- Level: Grad
Table of Contents:
Code:
[LIST]
[*] General Topology
[LIST]
[*] Sets
[LIST]
[*] Some Basic Terminology
[*] Denumerable Sets
[*] Zorn's Lemma
[/LIST]
[*] Topological Spaces
[LIST]
[*] Open and Closed Sets
[*] Connected Sets
[*] Compact Spaces
[*] Separation by Continuous Functions
[*] Exercises
[/LIST]
[*] Continuous Functions on Compact Sets
[LIST]
[*] The Stone-Weierstrass Theorem
[*] Ideals of Continuous Functions
[*] Ascoli's Theorem
[*] Exercises
[/LIST]
[/LIST]
[*] Banach and Hilbert Spaces
[LIST]
[*] Banach Spaces
[LIST]
[*] Definitions, the Dual Space, and the Hahn-Banach Theorem
[*] Banach Algebras
[*] The Linear Extension Theorem
[*] Completion of a Normed Vector Space
[*] Spaces with Operators
[LIST]
[*] Appendix: Convex Sets
[*] The Krein-Milman Theorem
[*] Mazur's Theorem
[/LIST]
[*] Exercises
[/LIST]
[*] Hilbert Space
[LIST]
[*] Hermitian Forms
[*] Functionals and Operators
[*] Exercises
[/LIST]
[/LIST]
[*] Integrations
[LIST]
[*] The General Integral
[LIST]
[*] 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 Integrals in R^p
[*] Exercises
[/LIST]
[*] Duality and Representation Theorems
[LIST]
[*] The Hilbert Space L^2(\mu)
[*] Duality Between L^1(\mu) and 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
[/LIST]
[*] Some Applications of Integration
[LIST]
[*] Convolution
[*] Continuity and Differentiation Under the Integral Sign
[*] Dirac Sequences
[*] The Schwartz Space and Fourier Transform
[*] The Fourier Inversion Formula
[*] The Poisson Summation Formula
[*] An Example of Fourier Transform Not in the Schwartz Space
[*] Exercises
[/LIST]
[*] Integration and Measures on Locally Compact Spaces
[LIST]
[*] Positive and Bounded Functionals on C_c(X)
[*] Positive Functionals as Integrals
[*] Regular Positive Measures
[*] Bounded Functionals as Integrals
[*] Localization of a Measure and of the Integral
[*] Product Measures on Locally Compact Spaces
[*] Exercises
[/LIST]
[*] Riemann-Stieltjes Integral and Measure
[LIST]
[*] Functions of Bounded Variation and the Stieltjes Integral
[*] Applications to Fourier Analysis
[*] Exercises
[/LIST]
[*] Distributions
[LIST]
[*] Definition and Examples
[*] Support and Localization
[*] Derivation of Distributions
[*] Distributions with Discrete Support
[/LIST]
[*] Integration on Locally Compact Groups
[LIST]
[*] Topological Groups
[*] The Haar Integrals, Uniqueness
[*] Existence of the Haar Integral
[*] Measures on Factor Groups and Homogeneous Spaces
[*] Exercises
[/LIST]
[/LIST]
[*] Calculus
[LIST]
[*] Differential Calculus
[LIST]
[*] Integration in One Variable
[*] The Derivative as Linear Map
[*] Properties of the Derivative
[*] Mean Value Theorem
[*] The Second Derivative
[*] Higher Derivatives and Taylor's Formula
[*] Partial Derivatives
[*] Differentiating Under the Integral Sign
[*] Differentiation of Sequences
[*] Exercises
[/LIST]
[*] Inverse Mappings and Differential Equations
[LIST]
[*] The Inverse Mapping Theorem
[*] The Implicit Mapping Theorem
[*] Existence Theorem of Differential Equations
[*] Local Dependence on Initial Conditions
[*] Global Smoothness of the Flow
[*] Exercises
[/LIST]
[/LIST]
[*] Functional Analysis
[LIST]
[*] The Open Mapping Theorem, Factor Spaces, and Duality
[LIST]
[*] The Open Mapping Theorem
[*] Orthogonality
[*] Applications of the Open Mapping Theorem
[/LIST]
[*] The Spectrum
[LIST]
[*] The Gelfand-Mazur Theorem
[*] The Gelfand Transform
[*] C*-Algebra
[*] Exercises
[/LIST]
[*] Compact and Fredholm Operators
[LIST]
[*] Compact Operators
[*] Fredholm Operators and the Index
[*] Spectral Theorem for Compact Operators
[*] Applications to Integral Equations
[*] Exercises
[/LIST]
[*] Spectral Theorem for Bounded Hermitian Operators
[LIST]
[*] 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
[/LIST]
[*] Further Spectral Theorems
[LIST]
[*] Projection Functions of Operators
[*] Self-Adjoint Operators
[*] Example: The Laplace Operators in the Plane
[/LIST]
[*] Spectral Measures
[LIST]
[*] 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
[*] Exercoses
[/LIST]
[/LIST]
[*] Global Analysis
[LIST]
[*] Local Integration of Differential Forms
[LIST]
[*] Sets of Measure 0
[*] Change of Variables Formula
[*] Differential Forms
[*] Inverse Image of a Form
[*] Appendix
[/LIST]
[*] Manifolds
[LIST]
[*] Atlases, Charts, Morphisms
[*] Submanifolds
[*] Tangent Spaces
[*] Partitions of Unity
[*] Manifolds with Boundary
[*] Vector Fields and Global Differential Equations
[/LIST]
[*] Integration and Measures on Manifolds
[LIST]
[*] 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
[/LIST]
[/LIST]
[*] Bibliography
[*] Table of Notation
[*] Index
[/LIST]
Last edited by a moderator: