Foundations of Real and Abstract Analysis

  • Douglas S. Bridges

Part of the Graduate Texts in Mathematics book series (GTM, volume 174)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Introduction

    1. Pages 1-8
  3. Real Analysis

    1. Front Matter
      Pages 9-9
  4. Abstract Analysis

    1. Front Matter
      Pages 123-123
    2. Pages 233-258
  5. Back Matter
    Pages 291-324

About this book


The core of this book, Chapters three through five, presents a course on metric, normed, and Hilbert spaces at the senior/graduate level. The motivation for each of these chapters is the generalisation of a particular attribute of the n Euclidean space R: in Chapter 3, that attribute is distance; in Chapter 4, length; and in Chapter 5, inner product. In addition to the standard topics that, arguably, should form part of the armoury of any graduate student in mathematics, physics, mathematical economics, theoretical statistics,. . . , this part of the book contains many results and exercises that are seldom found in texts on analysis at this level. Examples of the latter are Wong’s Theorem (3.3.12) showing that the Lebesgue covering property is equivalent to the uniform continuity property, and Motzkin’s result (5. 2. 2) that a nonempty closed subset of Euclidean space has the unique closest point property if and only if it is convex. The sad reality today is that, perceiving them as one of the harder parts of their mathematical studies, students contrive to avoid analysis courses at almost any cost, in particular that of their own educational and technical deprivation. Many universities have at times capitulated to the negative demand of students for analysis courses and have seriously watered down their expectations of students in that area. As a result, mathematics majors are graduating, sometimes with high honours, with little exposure to anything but a rudimentary course or two on real and complex analysis, often without even an introduction to the Lebesgue integral.


Hilbert space calculus differential equation functional analysis mathematical economics real analysis

Authors and affiliations

  • Douglas S. Bridges
    • 1
  1. 1.Department of MathematicsUniversity of WaikatoHamiltonNew Zealand

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag New York, Inc. 1998
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-98239-7
  • Online ISBN 978-0-387-22620-0
  • Series Print ISSN 0072-5285
  • Buy this book on publisher's site
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