Closed subspace of L(X,Y)

Let , be Normed Spaces with Complete. Then the compact Linear Operators form a closed subspace of .

Proof

Subspace: Must show that , , compact implies compact Given in , have subsequence with convergent to . And then has a subsequence with say. So . Closed: Let compact with . Need compact. Given : Choose with . Since is totally bounded, have for some whence , So Thus is totally bounded.

Note

Any limit of finite rank operators is compact.