# Banach Lecture 18

This lecture ( Banach18 ) was given on 10/7/09.

### Excerpts

Let $X,Y$ be Banach spaces. $T \in L(X,Y)$ Say $T$ is a Fredholm if $\dim Ker T < \infty$ and $\mbox{codim} TX < \infty$.  Note $TX$ is not assumed to be closed, the codimension is taken in the sense of ordinary linear algebra.

The index is then

$ind(T) = \dim \ker T - \mbox{Codim} TX$.

Prop. II.3.1 A Fredholm operator $T \in L(X,Y)$ maps subspaces of $X$ onto suspaces of $Y$.  In particular, $TX$ is closed.

Lemma II.3.1 Say $E,F \subset X$ are subspaces with $\dim F < \infty$.

1. $E + F is closed$
2. the image of $E$ in $X/F$ is closed.