*This thread originated from MSE: Compact Approximation*

*This is meant as lemma for: Approximation Property*

Given a Banach space $E$.

Denote compact domains by $\mathcal{C}$.

Denote compact operators by $\mathcal{C}(E)$.

Then there is a compact approximate identity: $$\forall C\in\mathcal{C}\,\exists C_N\in\mathcal{C}(E,E):\quad\|C_N-1\|_C:=\sup_{x\in C}\|C_Nx-x\|\stackrel{N\to\infty}\to0$$

How to construct such compact operators?

there existcompact operators $C_N$ such that this holds? $\endgroup$ – Nate Eldredge Feb 9 '15 at 18:26