# Prove the theorem

Please help me proof the theorem: If $A$ is compact operators, then

$a)$ (The Fredholm Alternative) $\sigma(A)=\{0\}\cup\Pi_{0}(A)$

$b)$ $\Pi_{0}(A)$ is either finite consist of a sequence converging to 0.

$c)$ The eigenspace $\{x:Ax=\lambda x\}$ is a finite-dimensional if $\lambda\neq0$.

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This proof is in every functional analysis textbook. Anyway, you should try to do it yourself first. – B0112358 Dec 20 '12 at 15:56