# Real and Functional Analysis by Lang

• Analysis
• micromass
In summary, "Real and Functional Analysis" by Serge Lang is a highly abstract and advanced textbook on real analysis. It covers topics such as general topology, continuous functions on compact sets, Banach and Hilbert spaces, integration and measures, distributions, calculus, functional analysis, and global analysis. While the book may be challenging for beginners, it offers valuable insights and knowledge in the field of real analysis. However, the author's writing style has been criticized for being harsh and difficult to follow at times.

## For those who have used this book

• ### Strongly Recommend

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• ### Lightly don't Recommend

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• ### Strongly don't Recommend

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• Total voters
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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:
this seems to be the same book that was originally issued as Analysis II by Lang. I used this book in a beginning grad class in real analysis in 1971 or so, but i really blew my well meaning class away. Do not do this. This is a very abstract and advanced version of real analysis and deserves to be a second or third version of the subject. (His version of integration theory is not for real valued or even complex valued functions, but for Banach space valued functions, helloooo?) I loved Lang in person and I respect and treasure some of his books, but as a textbook writer he could really be an arsehole (see if this spelling escapes the obscenity checker).

I liked it though. You learn some cool stuff here. Decades later when one of my colleagues said he was working on fredholm operators, but did not expect me to know what they were, thanks to lang I said, sure those are just Banach space operators with finite dimensional kernel and cokernel, right?

## 1. What is the main focus of "Real and Functional Analysis" by Lang?

The main focus of "Real and Functional Analysis" by Lang is to provide a comprehensive and rigorous treatment of real analysis and functional analysis. It covers topics such as topology, continuity, differentiability, integration, and metric spaces, as well as the theory of Banach and Hilbert spaces.

## 2. Is this book suitable for beginners in the field of analysis?

No, this book is not suitable for beginners as it assumes a strong background in mathematical analysis and advanced calculus. It is better suited for graduate students or researchers in mathematics.

## 3. How is the content organized in this book?

The book is divided into two parts: real analysis and functional analysis. The first part covers topics in one variable such as sequences, series, continuity, and differentiation. The second part focuses on advanced topics such as Banach and Hilbert spaces, linear operators, and spectral theory.

## 4. Does this book include exercises for practice?

Yes, this book contains a large number of exercises at the end of each chapter. These exercises range from basic to challenging and are designed to reinforce the concepts and techniques discussed in the text.

## 5. Can this book be used as a reference for researchers?

Yes, this book can be used as a reference for researchers as it covers a wide range of topics in real and functional analysis and provides a rigorous treatment of the subject. It also includes many examples and applications, making it a valuable resource for researchers in the field.

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