Foundations of Real and Abstract Analysis

1. Front Matter

   Pages i-xiv

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2. Introduction

   1. Introduction

      Pages 1-8

3. Real Analysis

   1. Front Matter

      Pages 9-9

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   2. Analysis on the Real Line

      Pages 11-77

   3. Differentiation and the Lebesgue Integral

      Pages 79-122

4. Abstract Analysis

   1. Front Matter

      Pages 123-123

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   2. Analysis in Metric Spaces

      Pages 125-171

   3. Analysis in Normed Linear Spaces

      Pages 173-231

   4. Hilbert Spaces

      Pages 233-258

   5. An Introduction to Functional Analysis

      Pages 259-290

5. Back Matter

   Pages 291-324

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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

• Department of Mathematics, University of Waikato, Hamilton, New Zealand

  Douglas S. Bridges

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