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Unformatted text preview: Analysis PhD Examination Sample Solutions 1. Let X and Y be normed linear spaces. Prove that the normed space B ( X, Y ) (comprising all bounded linear maps from X to Y ) is complete if and only if Y is complete. Solution ( ⇐ ) This is the ‘familiar’ direction. Let ( T n : n > 0) be Cauchy in B ( X, Y ). If x ∈ X then k T p ( x ) T q ( x ) k 6 k T p T q kk x k so that ( T n ( x ) : n > 0) is Cauchy in Y ; as Y is complete, we may define T ( x ) := lim n T n ( x ). Plainly, T : X → Y is linear. Let ε > 0 and choose N > 0 so that if p, q > N then k T p T q k 6 ε : if k x k 6 1 then k T p ( x ) T q ( x ) k 6 ε and passage to the limit as q → ∞ yields k T p ( x ) T ( x ) k 6 ε ; thus k T p T k 6 ε when p > N and so T p → T in B ( X, Y ). ( ⇒ ) Let ( y n : n > 0) be Cauchy in Y ; of course, we must pass to B ( X, Y ) by some means. One such: fix (0 6 =) a ∈ X and choose (how?) φ ∈ X * such that φ ( a ) = 1. For n > 0 define T n ∈ B ( X, Y ) by x ∈ X ⇒ T n ( x ) = φ ( x ) y n . Note that k T p ( x ) T q ( x ) k =  φ ( x ) k y p y q k 6 k y p y q kk φ kk x k so ( T n : n > 0) is Cauchy. Let T = lim n T n...
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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
 Robinson

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