Elementary Functional Analysis
In this introductory work on mathematical analysis, the noted mathematician Georgi E. Shilov begins with an extensive and important chapter on the basic structures of mathematical analysis: linear spaces, metric spaces, normed linear spaces, Hilbert spaces, and normed algebras. The standard models for all these spaces are sets of functions (hence the term "functional analysis"), rather than sets of points in a finite-dimensional space.
Chapter 2 is devoted to differential equations, and contains the basic theorems on existence and uniqueness of solutions of ordinary differential equations for functions taking values in a Banach space. The solution of the linear equation with constant (operator) coefficients is written in general form in terms of the exponential of the operator. This leads, in the finite-dimensional case, to explicit formulas not only for the solutions of first-order equations, but also to the solutions of higher-order equations and systems of equations. The third chapter presents a theory of curvature for curve in a multidimensional space.
The final two chapters essentially comprise an introduction to Fourier analysis. In the treatment of orthogonal expansions, a key role is played by Fourier series and the various kinds of convergence and summability for such series. The material on Fourier transforms, in addition to presenting the more familiar theory, also deals with problems in the complex domain, in particular with problems involving the Laplace transform.
Designed for students at the upper-undergraduate or graduate level, the text includes a set of problems for each chapter, with hints and answers at the end of the book.
1014424190
Elementary Functional Analysis
In this introductory work on mathematical analysis, the noted mathematician Georgi E. Shilov begins with an extensive and important chapter on the basic structures of mathematical analysis: linear spaces, metric spaces, normed linear spaces, Hilbert spaces, and normed algebras. The standard models for all these spaces are sets of functions (hence the term "functional analysis"), rather than sets of points in a finite-dimensional space.
Chapter 2 is devoted to differential equations, and contains the basic theorems on existence and uniqueness of solutions of ordinary differential equations for functions taking values in a Banach space. The solution of the linear equation with constant (operator) coefficients is written in general form in terms of the exponential of the operator. This leads, in the finite-dimensional case, to explicit formulas not only for the solutions of first-order equations, but also to the solutions of higher-order equations and systems of equations. The third chapter presents a theory of curvature for curve in a multidimensional space.
The final two chapters essentially comprise an introduction to Fourier analysis. In the treatment of orthogonal expansions, a key role is played by Fourier series and the various kinds of convergence and summability for such series. The material on Fourier transforms, in addition to presenting the more familiar theory, also deals with problems in the complex domain, in particular with problems involving the Laplace transform.
Designed for students at the upper-undergraduate or graduate level, the text includes a set of problems for each chapter, with hints and answers at the end of the book.
17.95 In Stock
Elementary Functional Analysis

Elementary Functional Analysis

by Georgi E. Shilov
Elementary Functional Analysis

Elementary Functional Analysis

by Georgi E. Shilov

eBook

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Overview

In this introductory work on mathematical analysis, the noted mathematician Georgi E. Shilov begins with an extensive and important chapter on the basic structures of mathematical analysis: linear spaces, metric spaces, normed linear spaces, Hilbert spaces, and normed algebras. The standard models for all these spaces are sets of functions (hence the term "functional analysis"), rather than sets of points in a finite-dimensional space.
Chapter 2 is devoted to differential equations, and contains the basic theorems on existence and uniqueness of solutions of ordinary differential equations for functions taking values in a Banach space. The solution of the linear equation with constant (operator) coefficients is written in general form in terms of the exponential of the operator. This leads, in the finite-dimensional case, to explicit formulas not only for the solutions of first-order equations, but also to the solutions of higher-order equations and systems of equations. The third chapter presents a theory of curvature for curve in a multidimensional space.
The final two chapters essentially comprise an introduction to Fourier analysis. In the treatment of orthogonal expansions, a key role is played by Fourier series and the various kinds of convergence and summability for such series. The material on Fourier transforms, in addition to presenting the more familiar theory, also deals with problems in the complex domain, in particular with problems involving the Laplace transform.
Designed for students at the upper-undergraduate or graduate level, the text includes a set of problems for each chapter, with hints and answers at the end of the book.

Product Details

ISBN-13: 9780486318684
Publisher: Dover Publications
Publication date: 03/18/2013
Series: Dover Books on Mathematics
Sold by: Barnes & Noble
Format: eBook
Pages: 352
File size: 13 MB
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About the Author

Soviet mathematician Georgi E. Shilov (1917-75) was a longtime Professor of Mathematics at Moscow State University and the author of several other successful Dover reprints.

Table of 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 of 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
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