# Banach Lecture 20

This lecture (banach20) was given on 10/12/09.

### Excerpts

Let $X$ be an inf. dim’l Banach space, and let $K(X) \subet \mathscr{L}(X)$ denote the ideal of compact operators.  Then the Calkin Algebra is defined to be $\mathscr{L}(X)/K(X)$.

• $T \in \mathscr{L}(X)$ is Fredholm iff its coset in the Calkin algebra is invertible.
• $sp_e(T)$ denotes the spectrum of the coset of $T$ in the Calkin algebra.

Prop II.3.2 if $T \in \masthscr{L}(X,Y)$ is Fredholm and $T^*$ is Fredholm and $ind(T^*) = -ind(T)$.

Rest of lecture has

1. def of left and right Fredholm.
2. Criterion for op. to be left or right Fredholm
3. Def on an inner product
4. Def of an inner product space
5. Results on properties of inner product space
6. Hilbert space: a complete inner product space.

Recall a topological space is separable if it contains a countable dense subset.