# Banach Lecture 9

This lecture ( banach9) was given on 9/16/09.

### Excerpts

A character on a topological group is a continuous homomorphism of the group into the group $\mathbb{T}$.  Most of the lecture was spent proving the following theorem.

Thm: The characters on $\mathbb{R}$ are the functions $x \mapsto \exp(ix t)$.

For $f \in L^1(\mathbb{R})$, let $\hat f (t) = \int_{\mathbb{R}} f(x) \exp(itx) dx$.  By the Riemann-Lesbegue lemma, $\widehat{L^1(\mathbb{R})} \subset C_0(\mathbb{R})$ but we do not have equality.