# Banach Lecture 13

This lecture (Banach13 ) was given on 9/25/09

### Excerpts

A subspace of Banach space is a closed vector subspace. is complemented in if there exists such that and are both subspaces; also say form a complimentary pair.

Lemma II.1.1 If is a subspace of finite codimenstion then is complimented in .

Lemma II.1.2 Let be a finite dimensional subspace. Then is complimented in .

Thm II.1.1 is complimented in iff for an idempotent.

Cor. if is complimented in then is complimented in .

comment: this is taken in the sense that any 1 – D subspace of determines a functional on .

Prop II.1.1 Let be Banach spaces and .

- is left invertible iff and is closed and complimented.
- is right invertible iff and is complimented in .

## About this entry

You’re currently reading “Banach Lecture 13,” an entry on Math Meandering

- Published:
- September 26, 2009 / 10:29 pm

- Category:
- Banach Algebras (Course)

- Tags:

## No comments yet

Jump to comment form | comment rss [?] | trackback uri [?]