245B_W09_hwk4

245B_W09_hwk4 - 245B, Winter 2009, Assignment 4: Notes and...

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245B, Winter 2009, Assignment 4: Notes and selected model answers (Model solutions follow question numbers in bold .) Folland Chapter 5 22. c. We assume from parts a. and b. that we have already defined the adjoint operator T L ( Y * ,X * ) associated to a bounded linear operator T L ( X,Y ) . We will prove that T is injective if and only if T has dense range. ( ) We prove the contrapositive. Suppose that R ( T ) $ Y . Then since R ( T ) is a proper closed subspace of Y , by the Hahn-Banach Theorem there exists some f Y * with f 6 = 0 but f | R ( T ) = 0 , and hence in particular f | R ( T ) = 0 . This latter restrictions tells us that for any x X we have T f ( x ) = def f ( Tx ) = 0 , and thus that T f = 0 , so T is not injective. ( ) Once again we prove the contrapositive. If T f = T g for some f 6 = g in Y * , then T h = 0 for h := f - g . This tells us that h ( Tx ) = T h ( x ) = 0 for all x X , so since h 6 = 0 we deduce that
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245B_W09_hwk4 - 245B, Winter 2009, Assignment 4: Notes and...

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