# Banach Lecture 13

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

### Excerpts

A subspace of Banach space is a closed vector subspace.  $E \subset X$ is complemented in $X$ if there exists $F \subset X$ such that $X = E \oplus F$ and $E,F$ are both subspaces; also say $E,F$ form a complimentary pair.

Lemma II.1.1 If $E \subset X$ is a subspace of finite codimenstion then $E$ is complimented in $X$.

Lemma II.1.2 Let $E \subset X$ be a finite dimensional subspace.  Then $E$ is complimented in $X$.

Thm II.1.1 $E \subset X$ is complimented in $X$ iff $E = P(X)$ for $P \in \mathscr{L}(X)$ an idempotent.

Cor. if $E$ is complimented in $X$ then $E^\perp$ is complimented in $X^*$.

comment: this is taken in the sense that any 1 – D subspace of $E^\perp$ determines a functional on $X$.

Prop II.1.1 Let $X,Y$ be Banach spaces and $T \in \mathscr{L}(X,Y)$.

1. $T$ is left invertible iff $\ker T = 0$ and $TX$ is closed and complimented.
2. $T$ is right invertible iff $TX = Y$ and $\ker T$ is complimented in $X$.