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Topological Groups: An Introduction

ISBN: 978-0-470-62451-7
367 pages
September 2010
Topological Groups: An Introduction (0470624515) cover image
A user-friendly introduction to metric and topological groups

Topological Groups: An Introduction provides a self-contained presentation with an emphasis on important families of topological groups. The book uniquely provides a modern and balanced presentation by using metric groups to present a substantive introduction to topics such as duality, while also shedding light on more general results for topological groups.

Filling the need for a broad and accessible introduction to the subject, the book begins with coverage of groups, metric spaces, and topological spaces before introducing topological groups. Since linear spaces, algebras, norms, and determinants are necessary tools for studying topological groups, their basic properties are developed in subsequent chapters. For concreteness, product topologies, quotient topologies, and compact-open topologies are first introduced as metric spaces before their open sets are characterized by topological properties. These metrics, along with invariant metrics, act as excellent stepping stones to the subsequent discussions of the following topics:

  • Matrix groups

  • Connectednesss of topological groups

  • Compact groups

  • Character groups

Exercises found throughout the book are designed so both novice and advanced readers will be able to work out solutions and move forward at their desired pace. All chapters include a variety of calculations, remarks, and elementary results, which are incorporated into the various examples and exercises.

Topological Groups: An Introduction is an excellent book for advanced undergraduate and graduate-level courses on the topic. The book also serves as a valuable resource for professionals working in the fields of mathematics, science, engineering, and physics.

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Preface.

1 Groups and Metrics.

1.1 Groups.

1.2 Metric and Topological Spaces.

1.3 Continuous Group Operations.

1.4 Subgroups and Their Quotient Spaces.

1.5 Compactness and Metric Groups.

2 Linear Spaces and Algebras.

2.1 Linear Structures on Groups.

2.2 Linear Functions.

2.3 Norms on Linear Spaces.

2.4 Continuous Linear Functions.

2.5 The Determinant Function.

3 The Subgroups of Rn.

3.1 Closed Subgroups.

3.2 Quotient Groups.

3.3 Dense Subgroups.

4 Matrix Groups.

4.1 General Linear Groups.

4.2 Orthogonal and Unitary Groups.

4.3 Triangular Groups.

4.4 One-Parameter Subgroups.

5 Connectedness of Topological Groups.

5.1 Connected Topological Spaces.

5.2 Connected Matrix Groups.

5.3 Compact Product Spaces.

5.4 Totally Disconnected Groups.

6 Metric Groups of Functions.

6.1 Real-Valued Functions.

6.2 The Compact-Open Topology.

6.3 Metric Groups of Isometries.

6.4 Metric Groups of Homeomorphisms.

6.5 Metric Groups of Homomorphisms.

7 Compact Groups.

7.1 Invariant Means.

7.2 Integral Equations.

7.3 Eigenfunctions.

7.4 Compact Abelian Groups.

7.5 Matrix Representations.

8 Character Groups.

8.1 Countable Discrete Abelian Groups.

8.2 The Duality Homomorphism.

8.3 Compactly Generated Abelian Groups.

8.4 A Duality Theorem.

Bibliography.

Index of Special Symbols.

Index.

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NELSON G. MARKLEY, PhD, was professor of mathematics at the University of Maryland for more than twenty-five years and also served as provost at Lehigh University. He has written numerous journal articles in the area of dynamical systems and is the author of Principles of Differential Equations, also published by Wiley.

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"Topological Groups: an Introduction is an excellent book for advanced undergraduate and graduate-level courses on the topic. The book also serves as a valuable resource for professional working in the fields of mathematics, science, engineering, and physics." (Bulletin Bibliographique, 2011)

"Recommended. Upper-division undergraduates through professionals." (Choice, 1 March 2011) 

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