Foundations of Real and Abstract Analysis

• Douglas S. Bridges
Book

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

1. Front Matter
Pages i-xiv

1. Pages 1-8
3. Real Analysis

1. Front Matter
Pages 9-9
2. Pages 11-77
3. Pages 79-122
4. Abstract Analysis

1. Front Matter
Pages 123-123
2. Pages 125-171
3. Pages 173-231
4. Pages 233-258
5. Pages 259-290
5. Back Matter
Pages 291-324

Introduction

The core of this book, Chapters 3 through 5, presents a course on metric, normed,andHilbertspacesatthesenior/graduatelevel. Themotivationfor each of these chapters is the generalisation of a particular attribute of the n Euclidean spaceR : in Chapter 3, that attribute isdistance; 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)showingthattheLebesguecoveringpropertyisequivalent 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 oftheirmathematicalstudies,studentscontrivetoavoidanalysiscoursesat 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 - jors 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.

Keywords

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 https://doi.org/10.1007/b97625
• Copyright Information Springer-Verlag New York, Inc. 1998
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-98239-7
• Online ISBN 978-0-387-22620-0
• Series Print ISSN 0072-5285