# Book:Georgi E. Shilov/Elementary Functional Analysis

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## Georgi E. Shilov:

## Georgi E. Shilov: *Elementary Functional Analysis*

Published $\text {1996}$, **Dover**

- ISBN 978-0486689234.

### Subject Matter

### Contents

**Preface**

**1. Basic Structures of Mathematical Analysis**

- 1.1 Linear Spaces

- 1.2 Metric Spaces

- 1.3 Normed Linear Spaces

- 1.4 Hilbert Spaces

- 1.5 Approximation on a Compactum

- 1.6 Differentiation and Integration in a Normed Linear Space

- 1.7 Continuous Linear Operators

- 1.8 Normed Algebras

- 1.9 Spectral Properties of Linear Operators

- Problems

**2. Differential Equations**

- 2.1 Definitions and Examples

- 2.2 The Fixed Point Theorem

- 2.3 Existence and Uniqueness Solutions

- 2.4 Systems of Equations

- 2.5 Higher-Order Equations

- 2.6 Linear Equations and Systems

- 2.7 The Homogeneous Linear Equation

- 2.8 The Nonhomogeneous Linear Equation

- Problems

**3. Space Curves**

- 3.1 Basic Concepts

- 3.2 Higher Derivatives

- 3.3 Curvature

- 3.4 The Moving Basis

- 3.5 The Natural Equations

- 3.6 Helices

- Problems

**4. Orthogonal Expansions**

- 4.1 Orthogonal Expansions in Hilbert Space

- 4.2 Trigonometric Fourier Series

- 4.3 Convergence of Fourier Series

- 4.4 Computations with Fourier Series

- 4.5 Divergent Fourier Series and Generalized Summation

- 4.6 Other Orthogonal Systems

- Problems

**5. The Fourier Transform**

- 5.1 The Fourier Integral and Its Inversion

- 5.2 Further Properties of the Fourier Transform

- 5.3 Examples and Applications

- 5.4 The Laplace Transform

- 5.5 Quasi-Analytic Classes of Functions

- Problems

**Hints and Answers**

**Bibliography**

**Index**