Introduction to Real Analysis

Introduction to Real Analysis

By William F. Trench
Book Description

Using a clear and informal approach, this book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.

This free edition is made available via Digital Commons @ Trinity in the hope that it will be useful as a textbook or reference.

Table of Contents
  • Trinity University
  • Digital Commons @ Trinity
    • 12-1-2013
  • Introduction to Real Analysis
    • William F. Trench
      • Recommended Citation
  • Table of Contents
  • Preface
  • Chapter 1 The Real Numbers
    • Section 1.1 The Real Number System
    • Section 1.2 Mathematical Induction
    • Section 1.3 The Real Line
  • Chapter 2 Differential Calculus of Functions of One Variable
    • Section 2.1 Functions and Limits
    • Section 2.2 Continuity
    • Differentiable Functions of One Variable
    • Section 2.4 L'Hospital's Rule
    • Section 2.5 Taylor's Theorem
  • Chapter 3 Integral Calculus of Functions of One
    • Section 3.1 Definition of the Integral
    • Section 3.2 Existence of the Integral
    • Section 3.3 Properties of the Integral
    • Section 3.4 Improper Integrals
    • Section 3.5 A More Advanced Look at the Existence of the Proper Riemann Integral
  • Chapter 4 Infinite Sequences and Series
    • Section 4.1 Sequences of Real Numbers
    • Section 4.2 Earlier Topics Revisited with Sequences
    • Section 4.3 Infinite Series of Constants
    • Section 4.4 Sequences and Series of Functions
    • Section 4.5 Power Series
  • Chapter 5 Real-Valued Functions of Several Varables
    • Section 5.1 Structure of R-.4n
    • Section 5.2 Continuous Real-Valued Functions of n Variables
    • Section 5.3 Partial Derivatives and the Differential
    • Section 5.4 The Chain Rule and Taylor's Theorem
  • Chapter 6 Vector-Valued Functions of Several Variables
    • Section 6.1 Linear Trensformations and Matrices
    • Section 6.2 Continuity and Differentiability of Transformationds
    • Section 6.3 The Inverse Function Theroem
    • Section 6.4 The Implicit Function Theorem
  • Chapter 7 Integrals of Functions of Several Variables
    • Section 7.1 Definition and Existence of the Multiple Integral
    • Section 7.2 Iterated Integrals and Mutiple Integrals
    • Section 7.3 Change of Variables in Multiple Integrals
  • Chapter 8 Metric Spaces
    • Section 8.1 Introduction to Metric Spaces
    • Section 8.2 Compact Sets in a Metric Space
    • Section 8.3 Continuous Functions on Metric Spaces
  • Answers to Selected
  • Index
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