Topological Groups: An Introduction

Written by an avid user of topological groups, this book provides a concrete introduction to metric and topological groups.  Important metric spaces are first introduced with the construction of a concrete metric, and a topological characterization of their open sets is immediately established.  Many proofs are presented in a purely topological fashion.  It is this use of topologies that affords the opportunity to discuss the limitations of some of the metric group results as well as the topological machinery needed to generalize them.  The book focuses on metric cases, thus keeping many proofs in familiar territory for readers new to the topic.  Readers learn how to replace the sequences with nets to obtain a general proof, and the result is an increased emphasis on topological groups and less on general topology.  The exercises have been designed to help hold the interest of advanced and beginning readers, and a variety of calculations, remarks, and necessary related facts have been placed within the exercises and are used throughout the text. When used in proofs, the calculations, remarks, and facts are carefully referred to so that advanced readers will know what is needed in the proof and will be able to move forward at a faster pace.   Novice readers will have to reference the exercise and workout a solution to become more engaged with the material.

Spis treści:
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 Gro ups.
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.