# Banach Lecture 15

This lecture ( lecture15) was given on 9/30/09.

### Excepts

A bunch of examples were given about compact operators.  For example, the Voltera operator on $L^p(0,1)$, $1\le p \le \infty$ defined by $Vf(x) = \int_0^x f(t)dt$.  The notes have a proof showing $V$ is compact for $p \ne 1$.

Prop II.2.2 An operator is compact iff its adjoint is compact.

Cor. $V$ is compact on $L^1(0,1)$ (because its adjoint is).

Prop. II.2.3 The range of a compact operator contains no infinite dimensional subspaces.