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.
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