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Some Modern Mathematics for Physicists and Other Outsiders
An Introduction to Algebra, Topology, and Functional Analysis
1st Edition - January 1, 1975
Author: Paul Roman
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
- Published: January 1, 1975
- Imprint: Pergamon
- eBook ISBN: 9781483187372