# Real and Functional Analysis by Lang

• Analysis

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## Main Question or Discussion Point

Code:
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[*] General Topology
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[*] Sets
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[*] Some Basic Terminology
[*] Denumerable Sets
[*] Zorn's Lemma
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[*] Topological Spaces
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[*] Open and Closed Sets
[*] Connected Sets
[*] Compact Spaces
[*] Separation by Continuous Functions
[*] Exercises
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[*] Continuous Functions on Compact Sets
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[*] The Stone-Weierstrass Theorem
[*] Ideals of Continuous Functions
[*] Ascoli's Theorem
[*] Exercises
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[*] Banach and Hilbert Spaces
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[*] Banach Spaces
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[*] Definitions, the Dual Space, and the Hahn-Banach Theorem
[*] Banach Algebras
[*] The Linear Extension Theorem
[*] Completion of a Normed Vector Space
[*] Spaces with Operators
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[*] Appendix: Convex Sets
[*] The Krein-Milman Theorem
[*] Mazur's Theorem
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[*] Exercises
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[*] Hilbert Space
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[*] Hermitian Forms
[*] Functionals and Operators
[*] Exercises
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[*] Integrations
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[*] The General Integral
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[*] 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
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[*] Duality and Representation Theorems
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[*] 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
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[*] Some Applications of Integration
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[*] 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
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[*] Integration and Measures on Locally Compact Spaces
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[*] 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
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[*] Riemann-Stieltjes Integral and Measure
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[*] Functions of Bounded Variation and the Stieltjes Integral
[*] Applications to Fourier Analysis
[*] Exercises
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[*] Distributions
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[*] Definition and Examples
[*] Support and Localization
[*] Derivation of Distributions
[*] Distributions with Discrete Support
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[*] Integration on Locally Compact Groups
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[*] Topological Groups
[*] The Haar Integrals, Uniqueness
[*] Existence of the Haar Integral
[*] Measures on Factor Groups and Homogeneous Spaces
[*] Exercises
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[*] Calculus
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[*] Differential Calculus
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[*] 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
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[*] Inverse Mappings and Differential Equations
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[*] The Inverse Mapping Theorem
[*] The Implicit Mapping Theorem
[*] Existence Theorem of Differential Equations
[*] Local Dependence on Initial Conditions
[*] Global Smoothness of the Flow
[*] Exercises
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[*] Functional Analysis
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[*] The Open Mapping Theorem, Factor Spaces, and Duality
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[*] The Open Mapping Theorem
[*] Orthogonality
[*] Applications of the Open Mapping Theorem
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[*] The Spectrum
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[*] The Gelfand-Mazur Theorem
[*] The Gelfand Transform
[*] C*-Algebra
[*] Exercises
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[*] Compact and Fredholm Operators
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[*] Compact Operators
[*] Fredholm Operators and the Index
[*] Spectral Theorem for Compact Operators
[*] Applications to Integral Equations
[*] Exercises
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[*] Spectral Theorem for Bounded Hermitian Operators
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[*] 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
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[*] Further Spectral Theorems
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[*] Projection Functions of Operators
[*] Example: The Laplace Operators in the Plane
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[*] Spectral Measures
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[*] 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
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[*] Global Analysis
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[*] Local Integration of Differential Forms
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[*] Sets of Measure 0
[*] Change of Variables Formula
[*] Differential Forms
[*] Inverse Image of a Form
[*] Appendix
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[*] Manifolds
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[*] Atlases, Charts, Morphisms
[*] Submanifolds
[*] Tangent Spaces
[*] Partitions of Unity
[*] Manifolds with Boundary
[*] Vector Fields and Global Differential Equations
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[*] Integration and Measures on Manifolds
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[*] 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
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[*] Bibliography
[*] Table of Notation
[*] Index
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