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.

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