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

Revival: Modern Analysis (1997)

ISBN 9781138560888
Published January 28, 2019 by CRC Press
584 Pages

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Book 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.
  • Table of Contents

    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
    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

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


    "A lucid and effective presentation of...sophisticated material"
    - Joseph A. Cima, Department of Mathematics, University of North Carolina, Chapel Hill
    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