Real Variables with Basic Metric Space Topology

✍ Scribed by Prof. Robert B. Ash

• Publisher: Robert B. Ash
• Year: 2007
• Tongue: English
• Leaves: 217
• Edition: Illustrated
• Category: Library

✦ Synopsis

Designed for a first course in real variables, this text presents the fundamentals for more advanced mathematical work, particularly in the areas of complex variables, measure theory, differential equations, functional analysis, and probability. Geared toward advanced undergraduate and graduate students of mathematics, it is also appropriate for students of engineering, physics, and economics who seek an understanding of real analysis.
The author encourages an intuitive approach to problem solving and offers concrete examples, diagrams, and geometric or physical interpretations of results. Detailed solutions to the problems appear within the text, making this volume ideal for independent study. Topics include metric spaces, Euclidean spaces and their basic topological properties, sequences and series of real numbers, continuous functions, differentiation, Riemann-Stieltjes integration, and uniform convergence and applications.

✦ Table of Contents

Title Page
Contents
Preface
1. Introduction
1.1 Basic Terminology
1.1.1 Definitions and Comments
1.2 Finite and Infinite Sets; Countably Infinite and Uncountably Infinite Sets
1.3 Distance and Convergence
1.3.1 Definitions and Comments
1.4 Minicourse in Basic Logic
1.4.1 Truth Tables
1.4.2 Types of Proof
1.4.3 Quantifiers
1.4.4 Mathematical Induction
1.4.5 Negations
1.5 Limit Points and Closure
2. Some Basic Topological Properties of R^p
2.1 Unions and Intersections of Open and Closed Sets
2.2 Compactness
2.2.1 Definition
2.2.2 Nested Set Property
2.2.3 Definition
2.2.5 Heine-Borel Theorem
2.3 Some Applications of Compactness
2.3.2 Bolzano-Weierstrass Theorem
2.4 Least Upper Bounds and Completeness
2.4.1 Definitions
2.4.4 Definitions and Comments
3. Upper and Lower Limits of Sequences of Real Numbers
3.1 Generalization of the Limit Concept
3.1.1 Definitions and Comments
3.2 Some Properties of Upper and Lower Limits
3.2.1 Definitions and Comments
3.2.4 Remark
3.3 Convergence of Power Series
4. Continuous Functions
4.1 Continuity: Ideas, Basic Terminology, Properties
4.1.1 Definition
4.1.4 Definition
4.1.7 Definition
4.2 Continuity and Compactness
4.2.3 Definition
4.3 Types of Discontinuities
4.3.2 Intermediate Value Theorem
4.4 The Cantor Set
4.4.2 Remarks
5. Differentiation
5.1 The Derivative and its Basic Properties
5.1.1 Definition and Comments
5.1.4 Mean Value Theorem
5.1.6 Generalized Mean Value Theorem
5.2 Additional Properties of The Derivative; Some Applications of the Mean Value Theorem
5.2.1 Intermediate Value Theorem for Derivatives
5.2.3 L'Hospital's Rule
5.2.4 Taylor's Formula with Remainder
6. Riemann-Stieltjes Integration
6.1 Definition of the Integral
6.1.1 Definitions and Comments
6.1.3 Definition
6.2 Properties of the Integral
6.2.3 Evaluation Formula
6.2.4 Fundamental Theorem of Calculus
6.3 Functions of Bounded Variation
6.3.1 Definitions and Comments
6.4 Some Useful Integration Theorems
6.4.1 Integration by Parts
6.4.2 Change of Variable Formula
6.4.3 Mean Value Theorem for Integrals
6.4.4 Upper Bounds on Integrals
6.4.5 Improper Integrals
7. Uniform Convergence and Applications
7.1 Pointwise and Uniform Convergence
7.1.1 Examples of Invalid Interchange of Operations
7.1.2 Definitions
7.1.3 Example
7.2 Uniform Convergence and Limit Operations
7.2.5 Dini's Theorem
7.3 The Weierstrass M-Test and Applications
7.3.1 Weierstrass M-Test
7.3.2 Example
7.3.3 An Everywhere Continuous, Nowhere Differentiate Function
7.4 Equicontinuity and the Arzela-Ascoli Theorem
7.4.3 Definition
7.4.4 Arzela-Ascoli Theorem
7.5 The Weierstrass Approximation Theorem
7.5.3 Weierstrass Approximation Theorem
8. Further Topological Results
8.1 The Extension Problem
8.1.3 Tietze Extension Theorem
8.2 Baire Category Theorem
8.2.1 Definitions and Comments
8.2.2 Baire Category Theorem
8.3 Connectedness
8.3.1 Definitions
8.4 Semicontinuous Functions
8.4.1 Definitions and Comments
9. Epilogue
9.1 Some Compactness Results
9.1.1 Definitions and Comments
9.2 Replacing Cantor's Nested Set Property
9.3 The Real Numbers Revisited
Solutions to Problems
Section 1
Section 2
Section 3
Section 4
Section 5
Section 6
Section 7
Section 8
Index

✦ Subjects

Real Analysis