Mathematical Analysis

Mathematical Analysis

A Special Course

1st Edition - January 1, 1965

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  • Author: G. Ye. Shilov
  • eBook ISBN: 9781483185378

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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.

Table of Contents

  • 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


    1. Further Remarks on Sets

    2. Theorems on Linear Functionals


    Other Titles in the Series

Product details

  • No. of pages: 498
  • Language: English
  • Copyright: © Pergamon 1965
  • Published: January 1, 1965
  • Imprint: Pergamon
  • eBook ISBN: 9781483185378

About the Author

G. Ye. Shilov

About the Editor

D. A. R. Wallace

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