分析入門

內(nèi)容概要

　　My 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）, my intent is to restore an intellectual liveliness to this course by offering the beginning student access to some truly significant achievements of the subject.

書籍目錄

Preface1 The Real Numbers1.1 Discussion： The Irrationality of 1.4141.2 Some Preliminaries1.3 The Axiom of Completeness1.4 Consequences of Completeness1.5 Cantor's Theorem1.6 Epilogue2 Sequences and Series2.1 Discussion： Rearrangements of Infinite Series2.2 The Limit of a Sequence2.3 The Algebraic and Order Limit Theorems2.4 The Monotone Convergence Theorem and a First Look at Infinite Series2.5 Subsequences and the Bolzano-Weierstrass Theorem2.6 The Cauchy Criterion2.7 Properties of Infinite Series2.8 Double Summations and Products of Infinite Series2.9 Epilogue3 Basic Topology of R3.1 Discussion： The Cantor Set3.2 Open and Closed Sets3.3 Compact Sets3.4 Perfect Sets and Connected Sets3.5 Baire's Theorem3.6 Epilogue4　Functional Limits and Continuity4.1 Discussion： Examples of Dirichlet and Thomae4.2 Functional Limits4.3 Combinations of Continuous Functions4.4 Continuous Functions on Compact Sets4.5 The Intermediate Value Theorem4.6 Sets of Discontinuity4.7 Epilogue5　The Derivative5.1 Discussion： Are Derivatives Continuous?5.2 Derivatives and the Intermediate Value Property5.3 The Mean Value Theorem5.4 A Continuous Nowhere-Differentiable FunCtion5.5 Epilogue6 Sequences and Series of Functions6.1 Discussion： Branching Processes6.2 Uniform Convergence of a Sequence of Functions6.3 Uniform Convergence and Differentiation6.4 Series of Functions6.5 Power Series6.6 Taylor Series6.7 Epilogue7 The Riemann Integral7.1 Discussion： How Should Integration be Defined?7.2 The Definition of the Riemann Integral7.3 Integrating Functions with Discontinuities7.4 Properties of the Integral7.5 The Fundamental Theorem of Calculus7.6 Lebesgue's Criterion for Riemann Integrability7.7 Epilogue8 Additional Topics8.1 The Generalized Riemann Integral8.2 Metric Spaces and the Baire Category Theorem8.3 Fourier Series8.4 A Construction of R From QBibliographyIndex

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