Spaces An Introduction to Real Analysis 1st edition by Tom Lindstrom ISBN 1470440628 9781470440626

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Product details:

ISBN 10:        1470440628
ISBN 13: 978-1470440626
Author:      Tom Lindstrom

 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.

Spaces An Introduction to Real Analysis 1st Table of contents:

Chapter 1: Foundations of Mathematical Logic

• 1.1. Proofs

• 1.2. Sets and Boolean Operations

• 1.3. Families of Sets

• 1.4. Functions

• 1.5. Relations and Partitions

• 1.6. Countability

• Notes and References for Chapter 1

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Chapter 2: The Foundation of Calculus

• 2.1. Epsilon-Delta and All That

• 2.2. Completeness

• 2.3. Four Important Theorems

• Notes and References for Chapter 2

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Chapter 3: Metric Spaces and Convergence

• 3.1. Definitions and Examples

• 3.2. Convergence and Continuity

• 3.3. Open and Closed Sets

• 3.4. Complete Spaces

• 3.5. Compact Sets

• 3.6. An Alternative Description of Compactness

• 3.7. The Completion of a Metric Space

• Notes and References for Chapter 3

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Chapter 4: Sequences, Series, and Function Spaces

• 4.1. Modes of Continuity

• 4.2. Modes of Convergence

• 4.3. Integrating and Differentiating Sequences

• 4.4. Applications to Power Series

• 4.5. Spaces of Bounded Functions

• 4.6. Spaces of Bounded, Continuous Functions

• 4.7. Applications to Differential Equations

• 4.8. Compact Sets of Continuous Functions

• 4.9. Differential Equations Revisited

• 4.10. Polynomials Are Dense in the Continuous Functions

• 4.11. The Stone-Weierstrass Theorem

• Notes and References for Chapter 4

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Chapter 5: Normed Spaces and Operators

• 5.1. Normed Spaces

• 5.2. Infinite Sums and Bases

• 5.3. Inner Product Spaces

• 5.4. Linear Operators

• 5.5. Inverse Operators and Neumann Series

• 5.6. Baire’s Category Theorem

• 5.7. A Group of Famous Theorems

• Notes and References for Chapter 5

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Chapter 6: Differential Calculus in Normed Spaces

• 6.1. The Derivative

• 6.2. Finding Derivatives

• 6.3. The Mean Value Theorem

• 6.4. The Riemann Integral

• 6.5. Taylor’s Formula

• 6.6. Partial Derivatives

• 6.7. The Inverse Function Theorem

• 6.8. The Implicit Function Theorem

• 6.9. Differential Equations Yet Again

• 6.10. Multilinear Maps

• 6.11. Higher Order Derivatives

• Notes and References for Chapter 6

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Chapter 7: Measure and Integration

• 7.1. Measure Spaces

• 7.2. Complete Measures

• 7.3. Measurable Functions

• 7.4. Integration of Simple Functions

• 7.5. Integrals of Nonnegative Functions

• 7.6. Integrable Functions

• 7.7. Spaces of Integrable Functions

• 7.8. Ways to Converge

• 7.9. Integration of Complex Functions

• Notes and References for Chapter 7

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Chapter 8: Constructing Measures

• 8.1. Outer Measure

• 8.2. Measurable Sets

• 8.3. Carathéodory’s Theorem

• 8.4. Lebesgue Measure on the Real Line

• 8.5. Approximation Results

• 8.6. The Coin Tossing Measure

• 8.7. Product Measures

• 8.8. Fubini’s Theorem

• Notes and References for Chapter 8

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Chapter 9: Fourier Series

• 9.1. Fourier Coefficients and Fourier Series

• 9.2. Convergence in Mean Square

• 9.3. The Dirichlet Kernel

• 9.4. The Fejér Kernel

• 9.5. The Riemann-Lebesgue Lemma

• 9.6. Dini’s Test

• 9.7. Pointwise Divergence of Fourier Series

• 9.8. Termwise Operations

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