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.

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