Modern Analysis / Edition 1

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Overview

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.

"...addresses real and abstract analysis...discusses measure theory, Lebesgue integration, and related topics... a sensible introduction to functional analysis."

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Editorial Reviews

Booknews
A textbook for a graduate course in real and abstract analysis or a basic course on measure theory. Assumes knowledge of normed vector space, the basic theorems in linear algebra, basic set theory and standard notation. Includes the Bochner integral that is widely used by engineers and others but often difficult to find in the literature, information on introductory topology to help readers understand the Reisz representation theory and related topics, various proofs of partial differential equations not often found, and exercises that illustrate and extend the theorems. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

  • ISBN-13: 9780849371660
  • Publisher: CRC Press
  • Publication date: 11/20/1997
  • Series: Studies in Advanced Mathematics Series , #26
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 592
  • Product dimensions: 6.41 (w) x 9.49 (h) x 1.29 (d)

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
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