Understanding Analysis - neues Buch

EAN (ISBN-13): 9780387215068
Erscheinungsjahr: 12
Herausgeber: Springer New York

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ISBN/EAN: 9780387215068

ISBN - alternative Schreibweisen:
978-0-387-21506-8
Alternative Schreibweisen und verwandte Suchbegriffe:
Autor des Buches: stephen abbott
Titel des Buches: understanding analysis

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Daten vom Verlag:

Autor/in: Stephen Abbott
Titel: Undergraduate Texts in Mathematics; Understanding Analysis
Verlag: Springer; Springer US
260 Seiten
Erscheinungsjahr: 2012-12-06
New York; NY; US
Sprache: Englisch
39,99 € (DE)
41,20 € (AT)
63,50 CHF (CH)
Available
XII, 260 p.

EA; E107; eBook; Nonbooks, PBS / Mathematik/Analysis; Reelle Analysis, Real-Variablen; Verstehen; Taylor series; algebra; boundary element method; convergence; differentiable function; function; functional; integral; integration; metric space; real analysis; real number; set; sets; variable; B; Real Functions; Mathematics and Statistics; BB

1 The Real Numbers.- 1.1 Discussion: The Irrationality of % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaSbdaGcaa % qaaiaaikdaaSqabaaaaa!3794! $$\\sqrt 2 $$.- 1.2 Some Preliminaries.- 1.3 The Axiom of Completeness.- 1.4 Consequences of Completeness.- 1.5 Cantor’s Theorem.- 1.6 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: TheCantor 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 Combinations of 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 Theorem.- 5.4 A Continuous Nowhere-Differentiable Function.- 5.5 Epilogue.- 6 Sequences and Series of Functions.- 6.1 Discussion: Branching Processes.- 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 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 Fourier Series.- 8.4 A Construction of R From Q.
Includes supplementary material: sn.pub/extras Request lecturer material: sn.pub/lecturer-material; A First Course in Real Analysis 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 philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Each chapter begins with the discussion of some motivating examples which concludes with a series of questions. The penultimate section of each chapter (the final section is a short epilogue) is written with the exercises incorporated into the exposition.