## Introduction to Analysis, 2/e

William R. Wade, University of Tennessee

Published July, 1999 by Prentice Hall Engineering/Science/Mathematics

Cloth
ISBN 0-13-014409-6

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Real Analysis-Mathematics

For one/two-semester, junior/senior-level courses in Advanced Calculus, Analysis I, Real Analysis taken by math majors. The first semester is usually a general requirement; the second semester is often an option for the more motivated students. Designed to challenge advanced students while bringing weaker students up to speed, this text is an advantageous alternative to most other analysis texts which either tend to be “too easy” (designed for an Intermediate Analysis course) or “too difficult” (designed for students headed for a Ph.D. in Pure Mathematics)—both of which usually tend to slight multidimensional material. Hailed for its readability, practicality, and flexibility, this text presents the Fundamental Theorems from a very practical point of view. Introduction to Analysis starts slowly and carefully, with a focused presentation of the material; saves extreme abstraction for the second semester; provides optional enrichment sections; includes many routine exercises and examples; and liberally supports (with examples and hints) what little theory is developed in the exercises.

NEW—Separate coverage of topology and analysis—Presents purely computational material first, in its own chapter (Ch. 8); then develops the topological material two alternate chapters (Ch. 9 on Euclidean spaces, and Ch. 9 on Metric spaces).

• Provides a much better flow, makes topology easier to understand.
NEW—Tightened level of rigor in the presentation of integers (Ch. 1)—Relegates all “well known” properties to the Appendices.
• Provides a shorter presentation and focuses the text even more on analysis.
NEW—Revised coverage on limits—Makes coverage of the concepts of limits supremum and infimum optional. Eliminates “limits through sets” and the concept of cluster point (except where it is essential in the metric space chapter). Relegates the concept sequentially compact to the exercises, where it has also been made optional.
NEW—Reorganized coverage of series—Separates series of constants and series of functions into two separate chapters. Includes many more routine exercises, and eliminates some of the more bizarre problems. Segregates several of the more esoteric tests (e.g., the log test, Raabe's test) into optional sections.
NEW—Many more elementary exercises—In the chapters on differentiability and integrability, especially in the multidimensional sections.
NEW—Consecutive numbering of Theorems, Definitions, Remarks, etc.
• Allows students and instructors to more easily find citations.
A uniform writing style and notation—Throughout. Presents material in small chunks (Remarks and Examples), with larger sections presenting an integrated point of view.
Unusually friendly, but rigorous treatment—Starts slowly (to give less capable students time to “get up to speed”), but contains all the major results associated with the standard material called “Advanced Calculus.” Uses analogy and geometry to motivate and explain the theory. Precedes many complicated proofs with a paragraph labeled “Strategy” which motivates the proof, shows why it was chosen, and why it should work. Follows many theorems with examples to show why each hypothesis is needed, allowing students to remember the hypotheses by recalling the examples. Presents proofs in complete detail, with each step carefully documented. Links proofs together in a way that teaches students to think ahead: “Why This Theorem?” Uses physical interpretations to examine some concepts from a second or third point of view.
• Meets students where they are, giving the weaker students all the support they need to develop basic requisite skills and knowledge, but not getting in the way of those who want a quicker pace and deeper focus. A practical focus—The text is honest and explicit about what is assumed from the very beginning.
• Students know what can be used, what proved.
Early introduction of the fundamental goals of analysis—And references to them often—e.g., interchanging two infinite operations (e.g., a limit sign and an integral sign), or examining how a limit operation interacts with an algebraic operation (e.g., set union or function inequality).
• Enables students to see that many separate theorems are part of a single point of view.
More coverage of multidimensional analysis than in most other texts—Provides an honest, geometrically motivated treatment of the calculus of several variables, which includes the Fundamental Theorems of Vector Calculus—a topic crucial to applied mathematicians that is usually glossed over in other texts or buried in an elegant algebraic presentation that is overwhelming to average undergraduates.
• Prepare students to study subfields like harmonic analysis and partial differential equations, where knowledge of vector calculus (especially, how actually to compute line and surface integrals instead of just throwing these terms around) is needed. These topics are in the forefront of contemporary research and are essential for students heading to graduate school in any quantitative field of study. Optional appendices and Enrichment sections.
• Enables serious students to delve further into the material and allows instructors to tailor their courses to their own tastes and student needs. Also keeps the main text from getting bogged down in esoterica (e.g., tests for convergence, partitions of unity, etc.).
An alternate chapter on metric spaces—Designed for instructors who feel these concepts are best presented first in generality, instead of in the concrete Euclidean space setting.
• Allows instructors to cover either chapter independently—without mentioning the other. All references to material from the topological chapter refers to BOTH these chapters, so students can skip either one of those chapters without loss of integrity.
Over 200 worked examples.
• Helps students review, shows them several different ways to apply a given result, or shows them the limitations of the theory (e.g., what happens to a result when one of the hypotheses is missing).
Over 600 exercises—Ranging from simple to challenging, including many applied-type exercises. Some are computational, some theoretical, some interpret the theory in a classical or practical way, some develop peripheral theory. Special boxed exercises develop theory that is used later in the book. Difficult exercises are presented in several steps—which serve as an outline to a solution.
• Allows students to test their comprehension of concepts, to focus clearly on the most important concepts/techniques, and to use the material in other contexts.

I. ONE-DIMENSIONAL THEORY.
1. The Real Number System.
2. Sequences in R.
3. Continuity on R.
4. Differentiability on R.
5. Integrability on R.
6. Infinite Series of Real Numbers.
7. Infinite Series of Functions.

II. MULTIDIMENSIONAL THEORY.
8. Euclidean Spaces.
9. Topology of Euclidean Spaces.
10. Metric Spaces.
11. Differentiability on Rn.
12. Integration on Rn.
13. Fundamental Theorems of Vector Calculus.
14. Fourier Series.
15. Differentiable Manifolds.
Appendices.
References.