Topological Groups: An Introduction

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.