Midterm2 - Analysis Midterm 2 Solutions Write solutions in...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Analysis Midterm 2 Solutions Write solutions in a neat and logical fashion, giving complete reasons for all steps. 1. Let Z be a closed subspace of the Banach space X . Prove that if X is reflexive then so is Z . Solution Recall that if T : Z X denotes inclusion then T ** Z = X T. Let Z ** . Passing to X ** and using the fact that X is reflexive, there exists x X such that T ** () = X ( x ). If X * vanishes on Z then ( check ) T * = 0 and therefore ( x ) = X x ( ) = T ** ( ) = ( T * ) = 0. The Hahn-Banach theorem now places z := x in Z (closed). We are almost done: T ** () = X ( z ) = T ** ( Z z ) so that = Z ( z ) by virtue of the fact that T ** is injective ( check ). 2. Let X be a normed space, let S, T : X X be linear maps and suppose that [ S, T ] : = ST- TS = I . Show that if n is a positive integer then [ S, T n ] = nT n- 1 . Deduce that S, T cannot be bounded....
View Full Document

This note was uploaded on 11/09/2011 for the course MAA 5228 taught by Professor Robinson during the Fall '09 term at University of Florida.

Page1 / 2

Midterm2 - Analysis Midterm 2 Solutions Write solutions in...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online