Real Analysis

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Paperback | 1 December 2000

At a Glance

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416 Pages

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25.4 x 17.78 x 2.16

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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.

Industry Reviews

'... extremely well written: very entertaining and motivating.' Adhemar Bultheel, Bulletin of the London Mathematical Society 'The author writes lucidly in a friendly, readable style and he is strong at motivating, anticipating and reviewing the various themes that permeate the text ... The overwhelming impression is that Real analysis was a labour of love for the author, written with a genuine reverence for both its beautiful subject matter and its creators, refiners and teachers down the ages. As such - and high praise indeed - it will sit very happily alongside classics such as Apostol's Mathematical analysis, Royden's Real analysis, Rudin's Real and complex analysis and Hewitt and Stromberg's Real and abstract analysis.' The Mathematical Gazette

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Real Analysis Probability & Statistics

Preface	p. xi
Metric Spaces
Calculus Review	p. 3
The Real Numbers	p. 3
Limits and Continuity	p. 14
Notes and Remarks	p. 17
Countable and Uncountable Sets	p. 18
Equivalence and Cardinality	p. 18
The Cantor Set	p. 25
Monotone Functions	p. 31
Notes and Remarks	p. 34
Metrics and Norms	p. 36
Metric Spaces	p. 37
Normed Vector Spaces	p. 39
More Inequalities	p. 43
Limits in Metric Spaces	p. 45
Notes and Remarks	p. 49
Open Sets and Closed Sets	p. 51
Open Sets	p. 51
Closed Sets	p. 53
The Relative Metric	p. 60
Notes and Remarks	p. 62
Continuity	p. 63
Continuous Functions	p. 63
Homeomorphisms	p. 69
The Space of Continuous Functions	p. 73
Notes and Remarks	p. 76
Connectedness	p. 78
Connected Sets	p. 78
Notes and Remarks	p. 87
Completeness	p. 89
Totally Bounded Sets	p. 89
Complete Metric Spaces	p. 92
Fixed Points	p. 97
Completions	p. 102
Notes and Remarks	p. 106
Compactness	p. 108
Compact Metric Spaces	p. 108
Uniform Continuity	p. 114
Equivalent Metrics	p. 120
Notes and Remarks	p. 126
Category	p. 128
Discontinuous Functions	p. 128
The Baire Category Theorem	p. 131
Notes and Remarks	p. 136
Function Spaces
Sequences of Functions	p. 139
Historical Background	p. 139
Pointwise and Uniform Convergence	p. 143
Interchanging Limits	p. 150
The Space of Bounded Functions	p. 153
Notes and Remarks	p. 160
The Space of Continuous Functions	p. 162
The Weierstrass Theorem	p. 162
Trigonometric Polynomials	p. 170
Infinitely Differentiable Functions	p. 176
Equicontinuity	p. 178
Continuity and Category	p. 183
Notes and Remarks	p. 185
The Stone-Weierstrass Theorem	p. 188
Algebras and Lattices	p. 188
The Stone-Weierstrass Theorem	p. 194
Notes and Remarks	p. 201
Functions of Bounded Variation	p. 202
Functions of Bounded Variation	p. 202
Helly's First Theorem	p. 210
Notes and Remarks	p. 212
The Riemann-Stieltjes Integral	p. 214
Weights and Measures	p. 214
The Riemann-Stieltjes Integral	p. 215
The Space of Integrable Functions	p. 221
Integrators of Bounded Variation	p. 225
The Riemann Integral	p. 232
The Riesz Representation Theorem	p. 234
Other Definitions, Other Properties	p. 239
Notes and Remarks	p. 242
Fourier Series	p. 244
Preliminaries	p. 244
Dirichlet's Formula	p. 250
Fejer's Theorem	p. 254
Complex Fourier Series	p. 257
Notes and Remarks	p. 258
Lebesgue Measure and Integration
Lebesgue Measure	p. 263
The Problem of Measure	p. 263
Lebesgue Outer Measure	p. 268
Riemann Integrability	p. 274
Measurable Sets	p. 277
The Structure of Measurable Sets	p. 283
A Nonmeasurable Set	p. 289
Other Definitions	p. 292
Notes and Remarks	p. 293
Measurable Functions	p. 296
Measurable Functions	p. 296
Extended Real-Valued Functions	p. 302
Sequences of Measurable Functions	p. 304
Approximation of Measurable Functions	p. 306
Notes and Remarks	p. 310
The Lebesgue Integral	p. 312
Simple Functions	p. 312
Nonnegative Functions	p. 314
The General Case	p. 322
Lebesgue's Dominated Convergence Theorem	p. 328
Approximation of Integrable Functions	p. 333
Notes and Remarks	p. 335
Additional Topics	p. 337
Convergence in Measure	p. 337
The L[subscript p] Spaces	p. 342
Approximation of L[subscript p] Functions	p. 350
More on Fourier Series	p. 352
Notes and Remarks	p. 356
Differentiation	p. 359
Lebesgue's Differentiation Theorem	p. 359
Absolute Continuity	p. 370
Notes and Remarks	p. 377
References	p. 379
Symbol Index	p. 395
Topic Index	p. 397
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Real Analysis

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

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