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.