Hello!

I'm going through the proof of the fact, that given a compact operator on a Banach space its spectrum is at most countable and 0 as the only possible accumulation point (which may or may not belong to the spectrum).

The reading has been fairly easy so far, but at the moment I'm somehow stuck at one step, namely:

The overall objective is to show that all points are isolated.

Let and be the restriction of to

There is a moment when it is evident that we need finiteness of .

So they argue that this follows from the fact, that is finite-dimensional.

I can't see at the moment how exactly it follows.

Does anybody have an idea?

P.S.:

I think I should mention that is chosen so that:

with , where .

Thanks in advance,

HAL