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5. Introduction to Real Analysis  # Introduction to Real Analysis, 2nd edition • Manfred Stoll

2nd edition

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## Overview

This text is a single variable real analysis text, designed for the one-year course at the junior, senior, or beginning graduate level. It provides a rigorous and comprehensive treatment of the theoretical concepts of analysis. The book contains most of the topics covered in a text of this nature, but it also includes many topics not normally encountered in comparable texts. These include the Riemann-Stieltjes integral, the Lebesgue integral, Fourier series, the Weiestrass approximation theorem, and an introduction to normal linear spaces.

KEY TOPICS: The Real Number System; Sequence Of Real Numbers; Structure Of Point Sets; Limits And Continuity; Differentiation; The Riemann And Riemann-Stieltjes Integral; Series of Real Numbers; Sequences And Series Of Functions; Orthogonal Functions And Fourier Series; Lebesgue Measure And Integration; Logic and Proofs; Propositions and Connectives

MARKET: For all readers interested in real analysis.

(Each chapter concludes with “Notes”, “Miscellaneous Exercises”, and a “Supplemental Reading”.)

1. The Real Number System.

Sets and Operations on Sets.

Functions.

Mathematical Induction.

The Least Upper Bound Property.

Consequences of the Least Upper Bound Property.

Binary and Ternary Expansions.

Countable and Uncountable Sets.

2. Sequence Of Real Numbers.

Convergent Sequences.

Limit Theorems.

Monotone Sequences.

Subsequences and the Bolzano-Weierstrass Theorem.

Limit Superior and Inferior of a Sequence.

Cauchy Sequences.

Series of Real Numbers.

3. Structure Of Point Sets.

Open and Closed Sets.

Compact Sets.

The Cantor Set.

4. Limits And Continuity.

Limit of a Function.

Continuous Functions.

Uniform Continuity.

Monotone Functions and Discontinuities.

5. Differentiation.

The Derivative.

The Mean Value Theorem.

L'Hôpital's Rule.

Newton's Method.

6. The Riemann And Riemann-Stieltjes Integral.

The Riemann Integral.

Properties of the Riemann Integral.

Fundamental Theorem of Calculus.

Improper Riemann Integrals.

The Riemann-Stieltjes Integral.

Numerical Methods.

Proof of Lebesgue's Theorem.

7. Series of Real Numbers.

Convergence Tests.

The Dirichlet Test.

Absolute and Conditional Convergence.

Square Summable Sequences.

8. Sequences And Series Of Functions.

Pointwise Convergence and Interchange of Limits.

Uniform Convergence.

Uniform Convergence and Continuity.

Uniform Convergence and Integration.

Uniform Convergence and Differentiation.

The Weierstrass Approximation Theorem.

Power Series Expansion.

The Gamma Function.

9. Orthogonal Functions And Fourier Series.

Orthogonal Functions.

Completeness and Parseval's Equality.

Trigonometric and Fourier Series.

Convergence in the Mean of Fourier Series.

Pointwise Convergence of Fourier Series.

10. Lebesgue Measure And Integration.

Introduction to Measure.

Measure of Open Sets; Compact Sets.

Inner and Outer Measure; Measurable Sets.

Properties of Measurable Sets.

Measurable Functions.

The Lebesgue Integral of a Bounded Function.

The General Lebesgue Integral.

Square Integrable Functions.

Appendix: Logic and Proofs.

Propositions and Connectives.

Rules of Inference.

Mathematical Proofs.

Use of Quantifiers.

Bibliography.

Hints and Solutions to Selected Exercises.

Notation Index.

Index.

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