Real analysis / N.L. Carothers.

Format:

Book

Author/Creator:

Carothers, N. L., 1952- author.

Language:

English

Subjects (All):

Mathematical analysis.

Physical Description:

1 online resource (xiii, 401 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 2000.

Language Note:

English

Summary:

This is a course in real analysis directed at advanced undergraduates and beginning graduate students in mathematics and related fields. Presupposing only a modest background in real analysis or advanced calculus, the book offers something to specialists and non-specialists. The course consists of three major topics: metric and normed linear spaces, function spaces, and Lebesgue measure and integration on the line. In an informal style, the author gives motivation and overview of new ideas, while supplying full details and proofs. He includes historical commentary, recommends articles for specialists and non-specialists, and provides exercises and suggestions for further study. This text for a first graduate course in real analysis was written to accommodate the heterogeneous audiences found at the masters level: students interested in pure and applied mathematics, statistics, education, engineering, and economics.

Contents:

Cover; Title; Copyright; Contents; Preface; PART ONE. METRIC SPACES; 1 Calculus Review; The Real Numbers; Limits and Continuity; Notes and Remarks; 2 Countable and Uncountable Sets; Equivalence and Cardinality; The Cantor Set; Monotone Functions; Notes and Remarks; 3 Metrics and Norms; Metric Spaces; Normed Vector Spaces; More Inequalities; Limits in Metric Spaces; Notes and Remarks; 4 Open Sets and Closed Sets; Open Sets; Closed Sets; The Relative Metric; Notes and Remarks; 5 Continuity; Continuous Functions; Homeomorphisms; The Space of Continuous Functions; Notes and Remarks

6 ConnectednessConnected Sets; Notes and Remarks; 7 Completeness; Totally Bounded Sets; Complete Metric Spaces; Fixed Points; Completions; Notes and Remarks; 8 Compactness; Compact Metric Spaces; Uniform Continuity; Equivalent Metrics; Notes and Remarks; 9 Category; Discontinuous Functions; The Baire Category Theorem; Notes and Remarks; PART TWO. FUNCTION SPACES; 10 Sequences of Functions; Historical Background; Pointwise and Uniform Convergence; Interchanging Limits; The Space of Bounded Functions; Notes and Remarks; 11 The Space of Continuous Functions; The Weierstrass Theorem

Trigonometric PolynomialsInfinitely Differentiable Functions; Equicontinuity; Continuity and Category; Notes and Remarks; 12 The Stone-Weierstrass Theorem; Algebras and Lattices; The Stone-Weierstrass Theorem; Notes and Remarks; 13 Functions of Bounded Variation; Functions of Bounded Variation; Helly's First Theorem; Notes and Remarks; 14 The Riemann-Stieltjes Integral; Weights and Measures; The Riemann-Stieltjes Integral; The Space of Integrable Functions; Integrators of Bounded Variation; The Riemann Integral; The Riesz Representation Theorem; Other Definitions, Other Properties

Notes and Remarks15 Fourier Series; Preliminaries; Dirichlet's Formula; Fejér's Theorem; Complex Fourier Series; Notes and Remarks; PART THREE. LEBESGUE MEASURE AND INTEGRATION; 16 Lebesgue Measure; The Problem of Measure; Lebesgue Outer Measure; Riemann Integrability; Measurable Sets; The Structure of Measurable Sets; A Nonmeasurable Set; Other Definitions; Notes and Remarks; 17 Measurable Functions; Measurable Functions; Extended Real-Valued Functions; Sequences of Measurable Functions; Approximation of Measurable Functions; Notes and Remarks; 18 The Lebesgue Integral; Simple Functions

Nonnegative FunctionsThe General Case; Lebesgue's Dominated Convergence Theorem; Approximation of Integrable Functions; Notes and Remarks; 19 Additional Topics; Convergence in Measure; The L[sub(p)] Spaces; Approximation of L[sub(p)] Functions; More on Fourier Series; Notes and Remarks; 20 Differentiation; Lebesgue's Differentiation Theorem; Absolute Continuity; Notes and Remarks; References; Symbol Index; Topic Index; A; B; C; D; E; F; G; H; I; J; K; L; M; N; O; P; Q; R; S; T; U; V; W; Y

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and indexes.

Other Format:

Print version:

ISBN:

9780511814228

1-316-08720-4

1-139-63598-0

1-139-64871-3

1-139-63826-2

1-139-64110-7

0-511-81422-4