**Pure and Applied Undergraduate Texts**

Volume: 29;
2017;
369 pp;
Hardcover

MSC: Primary 26; 28; 42; 46; 54;

**Print ISBN: 978-1-4704-4062-6
Product Code: AMSTEXT/29**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

**Electronic ISBN: 978-1-4704-4311-5
Product Code: AMSTEXT/29.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

#### Supplemental Materials

# Spaces: An Introduction to Real Analysis

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*Tom L. Lindstrøm*

Spaces is a modern introduction to real analysis at the advanced
undergraduate level. It is forward-looking in the sense that it first
and foremost aims to provide students with the concepts and techniques
they need in order to follow more advanced courses in mathematical
analysis and neighboring fields. The only prerequisites are a solid
understanding of calculus and linear algebra. Two introductory
chapters will help students with the transition from computation-based
calculus to theory-based analysis.

The main topics covered are metric
spaces, spaces of continuous functions, normed spaces, differentiation
in normed spaces, measure and integration theory, and Fourier
series. Although some of the topics are more advanced than what is
usually found in books of this level, care is taken to present the
material in a way that is suitable for the intended audience: concepts
are carefully introduced and motivated, and proofs are presented in
full detail. Applications to differential equations and Fourier
analysis are used to illustrate the power of the theory, and exercises
of all levels from routine to real challenges help students develop
their skills and understanding. The text has been tested in classes at
the University of Oslo over a number of years.

#### Readership

Undergraduate and graduate students interested in real analysis.

#### Reviews & Endorsements

[T]he presentation is done in a way to make the book eminently readable by undergraduate students...I think that reading 'Spaces' or taking a course based on the text would serve very well as a bridge between undergraduate level and modern graduate level mathematics.

-- Jason M. Graham, MAA Reviews

#### Table of Contents

# Table of Contents

## Spaces: An Introduction to Real Analysis

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Introduction –Mainly to the Students 114
- Chapter 1. Preliminaries: Proofs, Sets, and Functions 518
- Chapter 2. The Foundation of Calculus 2336
- Chapter 3. Metric Spaces 4356
- Chapter 4. Spaces of Continuous Functions 7992
- 4.1. Modes of continuity 7992
- 4.2. Modes of convergence 8194
- 4.3. Integrating and differentiating sequences 8699
- 4.4. Applications to power series 92105
- 4.5. Spaces of bounded functions 99112
- 4.6. Spaces of bounded, continuous functions 101114
- 4.7. Applications to differential equations 103116
- 4.8. Compact sets of continuous functions 107120
- 4.9. Differential equations revisited 112125
- 4.10. Polynomials are dense in the continuous function 116129
- 4.11. The Stone-Weierstrass Theorem 123136
- Notes and references for Chapter 4 131144

- Chapter 5. Normed Spaces and Linear Operators 133146
- Chapter 6. Differential Calculus in Normed Spaces 173186
- 6.1. The derivative 174187
- 6.2. Finding derivatives 182195
- 6.3. The Mean Value Theorem 187200
- 6.4. The Riemann Integral 190203
- 6.5. Taylor’s Formula 194207
- 6.6. Partial derivatives 201214
- 6.7. The Inverse Function Theorem 206219
- 6.8. The Implicit Function Theorem 212225
- 6.9. Differential equations yet again 216229
- 6.10. Multilinear maps 226239
- 6.11. Higher order derivatives 230243
- Notes and references for Chapter 6 238251

- Chapter 7. Measure and Integration 239252
- 7.1. Measure spaces 240253
- 7.2. Complete measures 248261
- 7.3. Measurable functions 252265
- 7.4. Integration of simple functions 257270
- 7.5. Integrals of nonnegative functions 262275
- 7.6. Integrable functions 271284
- 7.7. Spaces of integrable functions 276289
- 7.8. Ways to converge 285298
- 7.9. Integration of complex functions 288301
- Notes and references for Chapter 7 290303

- Chapter 8. Constructing Measures 291304
- 8.1. Outer measure 292305
- 8.2. Measurable sets 294307
- 8.3. Carathéodory’s Theorem 297310
- 8.4. Lebesgue measure on the real line 304317
- 8.5. Approximation results 307320
- 8.6. The coin tossing measure 311324
- 8.7. Product measures 313326
- 8.8. Fubini’s Theorem 316329
- Notes and references for Chapter 8 324337

- Chapter 9. Fourier Series 325338
- 9.1. Fourier coefficients and Fourier series 327340
- 9.2. Convergence in mean square 333346
- 9.3. The Dirichlet kernel 336349
- 9.4. The Fejér kernel 341354
- 9.5. The Riemann-Lebesgue Lemma 347360
- 9.6. Dini’s Test 350363
- 9.7. Pointwise divergence of Fourier series 354367
- 9.8. Termwise operations 356369
- Notes and references for Chapter 9 359372

- Bibliography 361374
- Index 363376
- Back Cover Back Cover1384