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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....
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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.
- Fall '09