Introduction

Harmonic Analysis is the study of the decomposition of representations of abstract algebraic structures acting on topological vector spaces.

Note: A table of the math symbols used below and their definitions is available in the Appendix.

• Foreword
• Old Introduction
• Manual of Style – How to read this wikibook

• The set theory notation and mathematical proofs, from the book Mathematical Proof
• The experience of working with calculus concepts, from the book Calculus

Part 1: General theory of Locally Compact Groups.

Topological Groups

• Exercises
  • Hints
  • Answers
• Topological Group - Definition and elementary properties.

Locally Compact Groups

• Locally Compact Groups - Definition and Elementary Properties

Banach Spaces of a Locally Compact Group

Haar Measure and ${\displaystyle L^{p}}$ spaces

The Group algebra and the Regular Representation

Square Integrable Representations

Representations of Compact Groups

The Group ${\displaystyle C^{*}}$-algebra and the Group Von Neumann algebra

Direct Integral of Representations

Characters of Locally Compact Groups

The Dual of a Locally Compact Group

Plancherel Theorem

Plancherel Measure

Topic 1: Fell Bundles

Part 2 Reductive Groups:

Semi-simple Lie Groups

Reductive Groups

Appendices

Here, you will find a list of unsorted chapters. Some of them listed here are highly advanced topics, while others are tools to aid you on your mathematical journey. Since this is the last heading for the wikibook, the necessary book endings are also located here.

• List of Mathematical Symbols
• List of Theorems
• References
• Index