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Understanding Analysis (Solution) written by Stephen Abbott
This book outlines an elementary, one-semester course which exposes students to both the process of rigor, and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination.
Author's primary goal in writing Understanding Analysis was to create an elementary one-semester book that exposes students to the rich rewards inherent in taking a mathematically rigorous approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it. There is a tendency, however, to center an introductory course too closely around the familiar theorems of the standard calculus sequence. Producing a rigorous argument that polynomials are continuous is good evidence for a well-chosen definition of continuity, but it is not the reason the subject was created and certainly not the reason it should be required study. By shifting the focus to topics where an untrained intuition is severely disadvantaged (e.g., rearrangements of infinite series, nowhere-differentiable continuous functions, Fourier series), author intent is to restore an intellectual liveliness to this course by offering the beginning student access to some truly significant achievements of the subject.
The author see three essential goals that a semester of real analysis should try to meet:
1. Students, especially those emerging from a reform approach to calculus, need to be convinced of the need for a more rigorous study of functions. The necessity of precise definitions and an axiomatic approach must be carefully motivated.
2. Having seen mainly graphical, numerical, or intuitive arguments, students need to learn what constitutes a rigorous mathematical proof and how to write one.
3. There needs to be significant reward for the difficult work of firming up the logical structure of limits. Specifically, real analysis should not be just an elaborate reworking of standard introductory calculus. Students should be exposed to the tantalizing complexities of the real line, to the subtleties of different flavors of convergence, and to the intellectual delights hidden in the paradoxes of the infinite.
Book Detail :-
Title: Understanding Analysis (Solution)
Author(s): Stephen Abbott
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About Author :-
Author Stephen Abbott is a professor of mathematics at Middlebury College, Middlebury, US and currently coeditor of Math Horizons.
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Book Contents :-
Understanding Analysis (Solution) written by Stephen Abbott cover the following topics.
1. The Real Numbers
2. Sequences and Series
3. Basic Topology of R
4. Functional Limits and Continuity
5. The Derivative
6. Sequences and Series of Functions
7. The Riemann Integral
8. Additional Topics
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