数学分析基础/Foundations of mathematical analysis

书籍目录

PrefacePreface to the Dover EditionI Sets and Functions  1.Sets  2.FunctionsII The Real Number System  3.The Algebraic Axioms of the Real Numbers  4.The Order Axiom of the Real Numbers  5.The Least-Upper-Bound Axiom  6.The Set of Positive Integers  7.Integers, Rationals, and ExponentsIII Set Equivalence  8.Definitions and Examples  9.Countable and Uncountable SetsIV Sequences of Real Numbers  10.Limit of a Sequence  11.Subsequences  12.The Algebra of Limits  13.Bounded Sequences  14.Further Limit Theorems  15.Divergent Sequences  16.Monotone Sequences and the Number e  17.Real Exponents  18.The Bolzano-Weierstrass Theorem  19.The Cauchy Condition  20.The lim sup and lim inf of Bounded Sequences  21.The lim sup and lim inf of Unbounded SequencesV Infinite Series  22.The Sum of an Infinite Series  23.Algebraic Operations on Series  24.Series with Nonnegative Terms  25.The Alternating Series Test  26.Absolute Convergence  27.Power Series  28.Conditional Convergence  29.Double Series and ApplicationsVI Limits of Real-Valued Functions and Continuous Functions on the Real Line  30.Definition of the Limit of a Function  31.Limit Theorems for Functions  32.One-Sided and Infinite Limits  33.Continuity  34.The Heine-Borel Theorem and a Consequence for Contihuous FunctionsVII Metric Spaces  35.The Distance Function  36.Rn, 12, and the Cauchy-Schwarz Inequality  37.Sequences in Metric Spaces  38.Closed Sets  39.Open Sets  40.Continuous Functions on Metric Spaces  41.The Relative Metric  42.Compact Metric Spaces  43.The Bolzano-Weierstrass Characterization of a Compact Metric Space  44.Continuous Functions on Compact Metric Spaces  45.Connected Metric Spaces  46.Complete Metric Spaces  47.Baire Category TheoremVIII Differential Calculus of the Real LineIX The Riemann-Stieltjes IntegralX Sequences and Series of FunctionsXI Transcendental FunctionsXII Inner Product Spaces and Fourier SeriesXIII Normed Linear Spaces and the Riesz Representation TheoremXIV The Lebesgue IntegralAppendix:Vector SpacesReferencesHints to Selected ExercisesIndexErrata

作者简介

This classroom-tested volume offers students of mathematics not only a well-defined view of the basics of modern analysis but also a broad spectrum of the ways in which analysis can be applied to sta-tistics, numerical analysis, Fourier series, differential equations,mathematical analysis, and functional analysis.    A self-contained textbook, it offers the background necessary for a firm grasp of the limit concept. (The first seven chapters could con-stitute a one-semester course on introduction to limits.) Subsequent chapters examine differential calculus of the real line, the Riemann-Stieltjes integral, sequences and series of functions, transcendental functions, inner product spaces and Fourier series, normed linear spaces and the Riesz representation theorem, and the Lebesgue inte-gral.Supplementary materials include an appendix on vector spaces and more than 750 exercises of varying degrees of difficulty (hints and solutions to selected exercises, indicated by an asterisk, appear at the back of the book).    Upper-level undergraduate students with a background in calculus will benefit from the teachings of this volume, as will beginning grad-uate students seeking a firm grounding in modern analysis.

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