Real Analysis

ISBN-10: 0521497566
ISBN-13: 9780521497565
Edition: 2000
Authors: N. L. Carothers
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Description: This text for a first graduate course in real analysis presupposes only a modest background knowledge. It consists of three major topics: metric and normed linear spaces, function spaces, and Lesbesgue measure and integration on the line.

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Book details

List price: $69.99
Copyright year: 2000
Publisher: Cambridge University Press
Publication date: 8/15/2000
Binding: Paperback
Pages: 416
Size: 7.00" wide x 9.75" long x 1.25" tall
Weight: 1.892
Language: English

This text for a first graduate course in real analysis presupposes only a modest background knowledge. It consists of three major topics: metric and normed linear spaces, function spaces, and Lesbesgue measure and integration on the line.

Preface
Metric Spaces
Calculus Review
The Real Numbers
Limits and Continuity
Notes and Remarks
Countable and Uncountable Sets
Equivalence and Cardinality
The Cantor Set
Monotone Functions
Notes and Remarks
Metrics and Norms
Metric Spaces
Normed Vector Spaces
More Inequalities
Limits in Metric Spaces
Notes and Remarks
Open Sets and Closed Sets
Open Sets
Closed Sets
The Relative Metric
Notes and Remarks
Continuity
Continuous Functions
Homeomorphisms
The Space of Continuous Functions
Notes and Remarks
Connectedness
Connected Sets
Notes and Remarks
Completeness
Totally Bounded Sets
Complete Metric Spaces
Fixed Points
Completions
Notes and Remarks
Compactness
Compact Metric Spaces
Uniform Continuity
Equivalent Metrics
Notes and Remarks
Category
Discontinuous Functions
The Baire Category Theorem
Notes and Remarks
Function Spaces
Sequences of Functions
Historical Background
Pointwise and Uniform Convergence
Interchanging Limits
The Space of Bounded Functions
Notes and Remarks
The Space of Continuous Functions
The Weierstrass Theorem
Trigonometric Polynomials
Infinitely Differentiable Functions
Equicontinuity
Continuity and Category
Notes and Remarks
The Stone-Weierstrass Theorem
Algebras and Lattices
The Stone-Weierstrass Theorem
Notes and Remarks
Functions of Bounded Variation
Functions of Bounded Variation
Helly's First Theorem
Notes and Remarks
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 Remarks
Fourier Series
Preliminaries
Dirichlet's Formula
Fejer's Theorem
Complex Fourier Series
Notes and Remarks
Lebesgue Measure and Integration
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
Measurable Functions
Measurable Functions
Extended Real-Valued Functions
Sequences of Measurable Functions
Approximation of Measurable Functions
Notes and Remarks
The Lebesgue Integral
Simple Functions
Nonnegative Functions
The General Case
Lebesgue's Dominated Convergence Theorem
Approximation of Integrable Functions
Notes and Remarks
Additional Topics
Convergence in Measure
The L[subscript p] Spaces
Approximation of L[subscript p] Functions
More on Fourier Series
Notes and Remarks
Differentiation
Lebesgue's Differentiation Theorem
Absolute Continuity
Notes and Remarks
References
Symbol Index
Topic Index

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