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## An Introduction to Algebra, Topology, and Functional Analysis

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Contents of Volume

Preface

Organization of the Book

Acknowledgments

Introduction

Part One: The Raw Materials of Mathematics

Chapter 1 Sets

1.1 Operations with Sets

1.2 Relations in Sets

Chapter 2 Maps

2.1 Composite Functions and Inverses

2.2 Equivalence Relations and Maps

2.3 Ordered Sets and Maps

2.4 Cardinal Numbers

2.5 Sequences and Families

Part Two: The Basic Structures of Mathematics

IIA : Algebraic Structures

Chapter 3 Algebraic Composition Laws and Systems

3.1 Morphisms of Algebraic Systems

Chapter 4 Survey of Special Algebraic Systems

4.1 Groups

4.2 Rings and Fields

4.3 Linear Spaces

4.4 Linear Algebras

4.5 Nonassociative Algebras

IIB : Topological Structures

Chapter 5 Topological Spaces

5.1 Examples; Metric Spaces

5.2 General Structure of Topological Spaces

5.3 Neighborhoods; Special Points; Closed Sets

5.4 Convergence

5.5 Continuity

5.6 Homeomorphism and Isometry

Chapter 6 Topological Spaces with Special Properties

6.1 Connected Spaces

6.2 Separable Spaces

6.3 Compact Spaces

6.4 Complete Metric Spaces

IIC: Measure Structures

Chapter 7 Measure Spaces

7.1 Measurable Spaces

7.2 Measure and Measure Spaces

Chapter 8 Theory of Integration

8.1 Measurable Functions

8.2 Definition of the Integral

8.3 General Properties of the Integral

8.4 Comments on Lebesgue and Lebesgue-Stieltjes Integrals

8.5 The Radon-Nikodym Theorem

Appendices

Appendix I Some Inequalities

Appendix III Annotated Reading List

Appendix IV Frequently Used Symbols

Index

- 1st Edition - May 9, 2014
- Author: Paul Roman
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 7 1 5 6 - 2
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 8 7 3 7 - 2

Some Modern Mathematics for Physicists and Other Outsiders: An Introduction to Algebra, Topology, and Functional Analysis, Volume 1 focuses on the operations, principles,… Read more

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Some Modern Mathematics for Physicists and Other Outsiders: An Introduction to Algebra, Topology, and Functional Analysis, Volume 1 focuses on the operations, principles, methodologies, and approaches employed in algebra, topology, and functional analysis. The publication first offers information on sets, maps, and algebraic composition laws and systems. Discussions focus on morphisms of algebraic systems, sequences and families, cardinal numbers, ordered sets and maps, equivalence relations and maps, composite functions and inverses, operations with sets, and relations in sets. The text then ponders on special algebraic systems, topological spaces, and topological spaces with special properties. Topics include complete metric spaces, compact spaces, separable and connected spaces, homeomorphism and isometry, convergence, continuity, general structure of topological spaces, rings and fields, linear spaces, linear algebras, and nonassociative algebras. The book elaborates on the theory of integration and measure spaces, including measurable spaces, general properties of the integral, and measureable functions. The publication is a valuable reference for theoretical physicists, research engineers, and scientists who are concerned with structural problems.

Contents of Volume

Preface

Organization of the Book

Acknowledgments

Introduction

Part One: The Raw Materials of Mathematics

Chapter 1 Sets

1.1 Operations with Sets

1.2 Relations in Sets

Chapter 2 Maps

2.1 Composite Functions and Inverses

2.2 Equivalence Relations and Maps

2.3 Ordered Sets and Maps

2.4 Cardinal Numbers

2.5 Sequences and Families

Part Two: The Basic Structures of Mathematics

IIA : Algebraic Structures

Chapter 3 Algebraic Composition Laws and Systems

3.1 Morphisms of Algebraic Systems

Chapter 4 Survey of Special Algebraic Systems

4.1 Groups

4.2 Rings and Fields

4.3 Linear Spaces

4.4 Linear Algebras

4.5 Nonassociative Algebras

IIB : Topological Structures

Chapter 5 Topological Spaces

5.1 Examples; Metric Spaces

5.2 General Structure of Topological Spaces

5.3 Neighborhoods; Special Points; Closed Sets

5.4 Convergence

5.5 Continuity

5.6 Homeomorphism and Isometry

Chapter 6 Topological Spaces with Special Properties

6.1 Connected Spaces

6.2 Separable Spaces

6.3 Compact Spaces

6.4 Complete Metric Spaces

IIC: Measure Structures

Chapter 7 Measure Spaces

7.1 Measurable Spaces

7.2 Measure and Measure Spaces

Chapter 8 Theory of Integration

8.1 Measurable Functions

8.2 Definition of the Integral

8.3 General Properties of the Integral

8.4 Comments on Lebesgue and Lebesgue-Stieltjes Integrals

8.5 The Radon-Nikodym Theorem

Appendices

Appendix I Some Inequalities

Appendix III Annotated Reading List

Appendix IV Frequently Used Symbols

Index

- No. of pages: 426
- Language: English
- Edition: 1
- Published: May 9, 2014
- Imprint: Pergamon
- Paperback ISBN: 9781483171562
- eBook ISBN: 9781483187372

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