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
Copyright 1997, 736 pp.
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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
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:
Explains the ideas behind developments and proofs
showing students that proofs come not from magical methods
but from natural processes, e.g.:
- 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).
Introduces concepts in stages, 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.
Features applications of abstract theorems to concrete
settings showing the power of an abstract approach in problem
- 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
Describes the interplay of various subjects
e.g., measure, variation, integration, and differentiation.
- Numerous applications to differential and integral equations,
infinite systems of linear equations, geometry/topology and existence
theorems of various sorts.
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.
Contains historical comments throughout which:
- Some extend the theory developed or present related
- 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
- 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.
6. Fubini's Theorem.
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.