This is meant as lemma for: Approximation Property
Given a Banach space $E$.
Denote compact operators by $\mathcal{C}(E)$.
Consider a compact domain $C\subseteq E$.
Then there is a compact approximate identity: $$C_N\in\mathcal{C}(E,E):\quad\|C_N-1\|_C:=\sup_{x\in C}\|C_Nx-x\|\stackrel{N\to\infty}\to0$$
How to construct such compact operators?
This is not always true!
Banach spaces which admit a compact approximation are said to possess the CAP.
(Enflo actually gave a counterexample for the CAP.)