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.