Real Analysis: Foundations

✍ Scribed by Sergei Ovchinnikov

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 187
• Series: Universitext
• Edition: 1
• Category: Library

✦ Synopsis

This textbook explores the foundations of real analysis using the framework of general ordered fields, demonstrating the multifaceted nature of the area. Focusing on the logical structure of real analysis, the definitions and interrelations between core concepts are illustrated with the use of numerous examples and counterexamples. Readers will learn of the equivalence between various theorems and the completeness property of the underlying ordered field. These equivalences emphasize the fundamental role of real numbers in analysis.

Comprising six chapters, the book opens with a rigorous presentation of the theories of rational and real numbers in the framework of ordered fields. This is followed by an accessible exploration of standard topics of elementary real analysis, including continuous functions, differentiation, integration, and infinite series. Readers will find this text conveniently self-contained, with three appendices included after the main text, covering an overview of natural numbers and integers, Dedekind's construction of real numbers, historical notes, and selected topics in algebra.

Real Analysis: Foundations is ideal for students at the upper-undergraduate or beginning graduate level who are interested in the logical underpinnings of real analysis. With over 130 exercises, it is suitable for a one-semester course on elementary real analysis, as well as independent study.

✦ Table of Contents

Preface
Contents
1 Rational Numbers
1.1 Definitions
1.2 Operations on Rational Numbers
1.3 Q as an Ordered Field
1.4 Limitations of Q
1.5 Convergence in an Ordered Field
Notes
Exercises
2 Real Numbers
2.1 Completeness Properties of Ordered Fields
2.2 Cauchy Completion of an Ordered Field
2.3 The Field R
2.4 Properties of the Field R
Notes
Exercises
3 Continuous Functions
3.1 Subsets of an Ordered Field
3.2 Continuity
3.3 Uniform Continuity
Notes
Exercises
4 Differentiation
4.1 Limits of Functions
4.2 The Derivative
4.3 Main Theorems
4.4 Convex Functions
Notes
Exercises
5 Integration
5.1 Partitions and Gaps
5.2 The Riemann Integral
5.3 Properties of the Riemann Integral
5.4 Step Functions
5.5 The Darboux Integral
5.6 Properties of Darboux Integrable Functions
Notes
Exercises
6 Infinite Series
6.1 Introduction
6.2 Series with Non-negative Terms
6.3 Alternating Series
6.4 Absolute Convergence
6.5 The Ratio Test
Notes
Exercises
A Natural Numbers and Integers
A.1 Natural Numbers
A.2 Integers
A.3 Rings
Notes
Exercises
B Dedekind's Construction of Real Numbers
B.1 Dedekind Cuts
B.2 Historical Notes
C A Panorama of Ordered Fields
C.1 Main Classes of Ordered Fields
C.2 The Field R((x))
C.3 The Field R(x)
Notes
References
Index

✦ Subjects

Real Analysis