# Banach Lecture 14

This lecture ( lecture14) was given on 9/28/09

### Excerpts

Let be a compact metric space and closed. The ideal of is the .

Thm: is complemeted in .

Then two examples were presented which show the converse of the corollary from last lecture doesn’t hold.

Another example, for , . Recall . It turns out this subspace is complemented in for and not complemented for .

### Compact Operators

, for Banach spaces, is compact if it maps bounded sets to relatively bounded sets (image has compact closure).

- if is linear and maps bounded sets to relatively compact sets, then is bounded
- if maps the unit ball in to a relatively bounded set in , then is compact.
- is compact iff for every bounded sequence , the sequence has a convergent subsequence.
- operators of finite rank are compact
- the identity operator on is compact iff is finite dimensional.
- The sum of two compact operators is compact.
- The product of a compact operator with a bounded operator (in either order) is compact; this implies an idempotent is compact iff its image has finite rank.

Prop. The set of compact operators in is closed in the operator norm.

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- Published:
- October 1, 2009 / 9:33 pm

- Category:
- Banach Algebras (Course)

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