Mathematical Analysis

A Special Course

1st Edition - January 1, 1965
Author: G. Ye. Shilov
Editor: D. A. R. Wallace
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 8 5 3 7 - 8

Mathematical Analysis: A Special Course covers the fundamentals, principles, and theories that make up mathematical analysis. The title first provides an account of set theory,… Read more

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Mathematical Analysis: A Special Course covers the fundamentals, principles, and theories that make up mathematical analysis. The title first provides an account of set theory, and then proceeds to detailing the elements of the theory of metric and normed linear spaces. Next, the selection covers the calculus of variations, along with the theory of Lebesgue integral. The text also tackles the geometry of Hilbert space and the relation between integration and differentiation. The last chapter of the title talks about the Fourier transform. The book will be of great use to individuals who want to expand and enhance their understanding of mathematical analysis.

Foreword

Chapter I. Sets

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 Functional

2. Extrema of Differentiable Functionals

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

4. Functionals of the Type b∫a (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

4. Measure of Sets and Theory of Lebesgue Integration

5. Generalisations

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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