# 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.

This article, or a section of it, needs explaining.In particular: Shilov died in 1975, so this is clearly a later edition, and very probably a translation. Research is invited as to exactly what edition is being referred to here.
"This Dover edition, first published in 1996, is an unabridged, slightly corrected republication of the work first published in English by The MIT Press, Cambridge, Massachusetts, 1974, as Volume 2 of the two-volume course "Mathematical Analysis" Hence this fact needs to be reflected in its filename or something.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

### 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**