# Book:Walter Rudin/Functional Analysis/Second Edition

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## Walter Rudin:

## Walter Rudin: *Functional Analysis (2nd Edition)*

Published $1991$, **McGraw-Hill**

- ISBN 978-0070619883.

### Subject Matter

### Contents

**Preface**

**Part I: General Theory**

**1 Topological Vector Spaces**

- Introduction
- Separation properties
- Linear mappings
- Finite-dimensional spaces
- Metrization
- Boundedness and continuity
- Seminorms and local convexity
- Quotient spaces
- Examples
- Exercises

**2: Completeness**

- Baire category
- The Banach-Steinhaus theorem
- The open mapping theorem
- The closed graph theorem
- Bilinear mappings
- Exercises

**3: Convexity**

- The Hahn-Banach theorems
- Weak topologies
- Compact convex sets
- Vector-valued integration
- Holomorphic functions
- Exercises

**4: Duality in Banach Spaces**

- The normed dual of a normed space
- Adjoints
- Compact operators
- Exercises

**5: Some applications**

- A continuity theorem
- Closed subspaces of Lp-spaces
- The range of a vector-valued measure
- A generalized Stone-Weierstrass theorem
- Two interpolation theorems
- Kakutani's fixed point theorem
- Haar measure on compact groups
- Uncomplemented subspaces
- Sums of Poisson kernels
- Two more fixed point theorems
- Exercises

**Part II: Distributions and Fourier Transforms**

**6: Test Functions and Distributions**

- Introduction
- Test functions spaces
- Calculus with distributions
- Localization
- Supports of distributions
- Distributions as derivatives
- Convolutions
- Exercises

**7: Fourier Transforms**

- Basic properties
- Tempered distributions
- Paley-Wiener theorems
- Sobolev's lemma
- Exercises

**8: Applications to Differential Equations**

- Fundamental solutions
- Elliptic functions
- Exercises

**9: Tauberian Theory**

- Wiener's theorem
- The prime number theorem
- The renewal equation
- Exercises

**Part III: Banach Algebras and Spectral Theory**

**10: Banach algebras**

- Introduction
- Complex homomorphisms
- Basic properties of spectra
- Symbolic calculus
- The group of invertible elements
- Lomonosov's invariant subspace theorem
- Exercises

**11: Commutative Banach Algebras**

- Ideals and Homomorhpisms
- Gelfand transforms
- Involutions
- Applications to noncommutative algebras
- Positive functionals
- Exercises

**12: Bounded Operators on a Hilbert Space**

- Basic facts
- Bounded operators
- A commutativity theorem
- Resolutions of the identity
- The spectral theorem
- Eigenvalues of normal operators
- Positive operators and square roots
- The group of invertible operators
- A characterization of B*-algebras
- An ergodic theorem
- Exercises

**13: Unbounded operators**

- Introduction
- Graphs and symmetric operators
- The Cayley transform
- Resolutions of the identity
- The spectral theorem
- Semigroups of operators
- Exercises

**Appendix A: Compactness and Continuity**

**Appendix B: Notes and Comments**

**Bibliography**

**List of Special symbols**

**Index**