Revival: Modern Analysis (1997): 1st Edition (Paperback) book cover

Revival: Modern Analysis (1997)

1st Edition

By Kenneth Kuttler

CRC Press

584 pages

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Description

Modern Analysis provides coverage of real and abstract analysis, offering a sensible introduction to functional analysis as well as a thorough discussion of measure theory, Lebesgue integration, and related topics. This significant study clearly and distinctively presents the teaching and research literature of graduate analysis:

  • Providing a fundamental, modern approach to measure theory

  • Investigating advanced material on the Bochner integral, geometric theory, and major theorems in Fourier Analysis Rn, including the theory of singular integrals and Milhin's theorem - material that does not appear in textbooks

  • Offering exceptionally concise and cardinal versions of all the main theorems about characteristic functions

  • Containing an original examination of sufficient statistics, based on the general theory of Radon measures

    With an ambitious scope, this resource unifies various topics into one volume succinctly and completely. The contents span basic measure theory in an abstract and concrete form, material on classic linear functional analysis, probability, and some major results used in the theory of partial differential equations. Two different proofs of the central limit theorem are examined as well as a straightforward approach to conditional probability and expectation.

    Modern Analysis provides ample and well-constructed exercises and examples. Introductory topology is included to help the reader understand such items as the Riesz theorem, detailing its proofs and statements. This work will help readers apply measure theory to probability theory, guiding them to understand the theorems rather than merely follow directions.

  • Reviews

    "A lucid and effective presentation of…sophisticated material"

    - Joseph A. Cima, Department of Mathematics, University of North Carolina, Chapel Hill

    "Ambitious…Conversant…Great"

    Steven G. Krantz, Department of Math, Washington University, St. Louis, Missouri

    "The author has chosen his topics well to demonstrate how even the oldest subjects can have a "modern" approach that improves on the original. The text is all business and very readable, especially for the mathematically prepared reader. There is a lot to recommend from the use of this book, not the least of which is the fact that the reader participates in the continuing documentation of "modern" mathematical analysis."

    -Timothy Hall, PQI Consulting

    Table of Contents

    Preface

    Set Theory and General Topology

    Compactness and Continuous Functions

    Banach Spaces

    Hilbert Spaces

    Calculus in Banach Space

    Locally Convex Topological Vector Spaces

    Measures and Measurable Functions

    The Abstract Lebesgue Integral

    The Construction of Measures

    Lebesgue Measure

    Product Measure

    The Lp Spaces

    Representation Theorems

    Fundamental Theorem of Calculus

    General Radon Measures

    Fourier Transforms

    Probability

    Weak Derivatives

    Hausdorff Measures

    The Area Formula

    The Coarea Formula

    Fourier Analysis in Rn

    Integration for Vector Valued Functions

    Convex Functions

    Appendix 1: The Hausdorff Maximal Theorem

    Appendix 2: Stone's Theorem and Partitions of Unity

    Appendix 3: Taylor Series and Analytic Functions

    Appendix 4: The Brouwer Fixed Point Theorem

    References

    Index

    About the Author/Editor

    Kenneth L.Kuttler,Jr. is a Professor at Department of Math, Brigham Young University

    About the Series

    CRC Press Revivals

    Learn more…

    Subject Categories

    BISAC Subject Codes/Headings:
    MAT000000
    MATHEMATICS / General
    MAT003000
    MATHEMATICS / Applied
    MAT037000
    MATHEMATICS / Functional Analysis