Understanding Analysis 2nd edition by Stephen Abbott – Ebook PDF Instant Download/Delivery: 1493927128 , 9781493927128
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ISBN 10: 1493927128
ISBN 13: 9781493927128
Author: Stephen Abbott
This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one. Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler’s computation of ζ(2), the Weierstrass Approximation Theorem, and the gamma function are now among the book’s cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.
Understanding Analysis 2nd Table of contents:
1 The Real Numbers
1.1 Discussion: The Irrationality of 2
1.2 Some Preliminaries
1.3 The Axiom of Completeness
1.4 Consequences of Completeness
1.5 Cardinality
1.6 Cantor’s Theorem
1.7 Epilogue
2 Sequences and Series
2.1 Discussion: Rearrangements of Infinite Series
2.2 The Limit of a Sequence
2.3 The Algebraic and Order Limit Theorems
2.4 The Monotone Convergence Theorem and a First Look at Infinite Series
2.5 Subsequences and the Bolzano–Weierstrass Theorem
2.6 The Cauchy Criterion
2.7 Properties of Infinite Series
2.8 Double Summations and Products of Infinite Series
2.9 Epilogue
3 Basic Topology of R
3.1 Discussion: The Cantor Set
3.2 Open and Closed Sets
3.3 Compact Sets
3.4 Perfect Sets and Connected Sets
3.5 Baire’s Theorem
3.6 Epilogue
4 Functional Limits and Continuity
4.1 Discussion: Examples of Dirichlet and Thomae
4.2 Functional Limits
4.3 Continuous Functions
4.4 Continuous Functions on Compact Sets
4.5 The Intermediate Value Theorem
4.6 Sets of Discontinuity
4.7 Epilogue
5 The Derivative
5.1 Discussion: Are Derivatives Continuous?
5.2 Derivatives and the Intermediate Value Property
5.3 The Mean Value Theorems
5.4 A Continuous Nowhere-Differentiable Function
5.5 Epilogue
6 Sequences and Series of Functions
6.1 Discussion: The Power of Power Series
6.2 Uniform Convergence of a Sequence of Functions
6.3 Uniform Convergence and Differentiation
6.4 Series of Functions
6.5 Power Series
6.6 Taylor Series
6.7 The Weierstrass Approximation Theorem
6.8 Epilogue
7 The Riemann Integral
7.1 Discussion: How Should Integration be Defined?
7.2 The Definition of the Riemann Integral
7.3 Integrating Functions with Discontinuities
7.4 Properties of the Integral
7.5 The Fundamental Theorem of Calculus
7.6 Lebesgue’s Criterion for Riemann Integrability
7.7 Epilogue
8 Additional Topics
8.1 The Generalized Riemann Integral
8.2 Metric Spaces and the Baire Category Theorem
8.3 Euler’s Sum
8.4 Inventing the Factorial Function
8.5 Fourier Series
8.6 A Construction of R From Q
Bibliography
Index
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