## Real Analysis, 1/e

Andrew M. Bruckner, University of California, Santa Barbara
Judith B. Bruckner
Brian Thomson, Simon Fraser University

Published September, 1996 by Prentice Hall Engineering/Science/Mathematics

Cloth
ISBN 0-13-458886-X

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Real Analysis-Mathematics

An important new graduate text that motivates the reader by providing the historical evolution of modern analysis. Sensitive to the needs of students with varied backgrounds and objectives, this text presents the tools, methods and history of analysis. The text includes the topics that “every graduate student must know” as well as specialized topics that prepare students for further study in analysis. User-friendly in approach, it includes numerous examples and exercises to illustrate and motivate the concepts.

An introductory chapter provides background material as well as a mini-overview of much of the course, making the text accessible to students with varied backgrounds — concepts introduced here help motivate ideas that reappear later in a more abstract setting.
Uses a wealth of examples to introduce topics and to illustrate important concepts:

• Some examples are reintroduced in later sections so students can consider how the new material affects familiar examples (e.g., information on the Fredholm operator is gleaned from abstract theorems about metric spaces, operators, and Hilbert spaces).
Explains the ideas behind developments and proofs — showing students that proofs come not from “magical methods” but from natural processes, e.g.:
• Introduces the Vitali Covering Theorem by first trying to establish a generalization of a simple “growth” condition known to all students; shows that a natural attempt at establishing this generalization has difficulties and how the VCT is exactly what's needed to overcome them.
• Introduces the Radon-Nikodym Theorem by first discussing briefly how the proof might look for the concrete case of Lebesgue-Stieltjes measures; later in Chapter 8 shows students that one can interpret dgif/pi.gifn/dgif/pi.gifm as a limit of quotients of measures in a natural way; carries out the details first for R^n and then for abstract spaces, and shows that such pointwide derivatives arise in settings familiar to students.
• Approaches the Baire Category Theory by using the Banach-Mazur game to help students see that the concept of “nowhere dense” provides a natural sense in which a set can be small.
Introduces concepts in stages, e.g.:
• Mass distributions Lebesgue-Stieltjes measures — introduces additive set functions as mass distributions, but shows limitations in their scope (Section 2.2). Then, after developing a bit of measure theory, it is shown that countable additivity removes the flaw (Exercise 2:4.10). Further examples and exercises show how application of the general theory results in a satisfying development (Exercise 2:11.2). This early material prepares students for the formal development of Lebesgue-Stieltjes measures later (Sections 3.5 and 3.6).
Features applications of abstract theorems to concrete settings — showing the power of an abstract approach in problem solving, e.g.:
• Numerous applications to differential and integral equations, infinite systems of linear equations, geometry/topology and existence theorems of various sorts.
Describes the interplay of various subjects — e.g., measure, variation, integration, and differentiation.
Each section includes exercises that use the examples of that section, as well as additional examples; an extensive list of problems at the end of each chapter.
• Some extend the theory developed or present related material.
• Others provide some interesting and revealing (but not necessarily well-known) examples.
• Some can form the basis of projects.
• Describe the evolution of various concepts over time.
• Show some obstacles that impeded progress and how they were overcome.
• Lend Perspective to the more recent formulations of classical results. (e.g., development of the integral).

1. Background and Preview.
2. Measure Spaces.
3. Metric Outer Measures.
4. Measurable Functions.
5. Integration.
6. Fubini's Theorem.
7. Differentiation.
8. Differentiation of Measures.
9. Metric Spaces.
10. Baire Category.
11. Analytic Sets.
12. Banach Spaces.
13. The Lp Spaces
14. Hilbert Spaces.
15. Fourier Series.
Index.