Compact Linear Map

Let , be Normed Spaces.
Say Linear is compact if is Compact.

Notes

• If compact, then certainly, is bounded, so is a Linear Operator.
• For Complete, is compact iff Totally bounded
• is compact
  if and only if
  For any in ,
  there is a subsequence such that converges in .

Proposition

, , normed, and .
Then:

1. If is compact is compact
2. If is compact is compact

Proof

1. Given in :
   There is a subsequencce with convergent so convergent ( is continuous)
2. Given in :
   have bounded, so there is a subsequence with convergent because is compact.