Mathematical Analysis

A Special Course

1st Edition - January 1, 1965
Author: G. Ye. Shilov
Editors: I.N. Sneddon, M. Stark, K.A.H. Gravett
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 3 9 1 3 - 5

Mathematical Analysis: A Special Course focuses on the study of mathematical analysis. The book first discusses set theory, including operations on sets, countable sets,… Read more

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Mathematical Analysis: A Special Course focuses on the study of mathematical analysis. The book first discusses set theory, including operations on sets, countable sets, equivalence of sets, and sets of the power of the continuum. The text also discusses the elements of the theory of metric and normed linear spaces. Topics include convergent sequences and closed sets; theorem of the fixed point; normed linear spaces; and continuous functions and compact spaces. The selection also discusses the calculus of variations; the theory of the integral; and geometry of Hilbert space. The text also covers differentiation and integration, including functions of bounded variation, derivative of a non-decreasing function, differentiation of functions of sets, and the Stieltjes integral. The book also looks at the Fourier transform. Topics include convergence of Fourier series; Laplace transform; Fourier transform in the case of various independent variables; and quasi-analytic classes of functions. The text is a valuable source of data for readers interested in the study of mathematical analysis.

Contents

Foreword

Chapter I. Sets 1

1. Sets, Subsets, Inclusions

2. Operations On Sets

3. Equivalence of Sets

4. Countable Sets

5. Sets of the Power of The Continuum

6. Sets of Higher Powers

Chapter II. Metric Spaces

1. Definition and Examples of Metric Spaces. Isometry

2. Open Sets

3. Convergent Sequences and Closed Sets

4. Complete Spaces

5. Theorem of The Fixed Point

6. Completion of A Metric Space

7. Continuous Functions and Compact Spaces

8. Normed Linear Spaces

9. Linear and Quadratic Functions On A Linear Space

Chapter III. The Calculus of Variations

1. Differentiable Functionals

2. Extrema of Differentiable Functionals

3. Functionals of the Type b∫a f(x, y, y') dx

4. Functionals of The Type b∫a f(x, y, y') dx (Continued)

5. Functionals with Several Unknown Functions

6. Functionals with Several Independent Variables

7. Functionals with Higher Derivatives

Chapter IV. Theory of the Integral

1. Sets of Measure Zero and Measurable Functions

2. The Class C+

3. Summable Functions 156

4. Measure of Sets and Theory of Lebesgue Integration

5. Generalizations

Chapter V. Geometry of Hilbert Space

1. Basic Definitions and Examples

2. Orthogonal Resolutions

3. Linear Operators

4. Integral Operators with Square-Summable Kernels

5. The Sturm-Liouville Problem

6. Non-Homogeneous Integral Equations with Symmetric Kernels

7. Non-Homogeneous Integral Equations with Arbitrary Kernels

8. Applications to Potential Theory

9. Integral Equations with Complex Parameters

Chapter VI. Differentiation and Integration

1. Derivative of A Non-Decreasing Function

2. Functions of Bounded Variation

3. Determination of A Function From Its Derivative

4. Functions of Several Variables

5. The Stieltjes Integral

6. The Stieltjes Integral (Continued)

7. Applications of The Stieltjes Integral In Analysis

8. Differentiation of Functions of Sets

Chapter VII. The Fourier Transform

1. On The Convergence of Fourier Series

2. The Fourier Transform

3. The Fourier Transform (Continued)

4. The Laplace Transform

5. Quasi-Analytic Classes of Functions

6. The Fourier Transform In The Class L2(— ∞,∞)

7. The Fourier-Stieltjes Transform

8. The Fourier Transform In The Case of Several Independent Variables

Supplement

1. Further Remarks on Sets

2. Theorems On Linear Functionals

Index

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