Sample - x | Ay i ; prove that it is automatically...

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Analysis PhD Examination Sample Answer SIX questions. Write solutions in a neat and logical fashion, giving complete reasons for all steps and stating carefully any substantial theorems used. 1. Let X be a normed space, Z X a closed subspace and u X ± Z . Prove that the subspace of X spanned by Z and u is closed in X . 2. Prove that the spaces c (of all convergent scalar sequences) and c 0 (of all null scalar sequences) are not isometrically isomorphic when equipped with the sup norm. 3. Let X be a Banach space and T : X X a linear map such that T T = T . Prove that T is continuous iff Ker T and Ran T are closed. 4. Let X and Y are closed subspaces of a Hilbert space. Prove that if X Y then X + Y is closed. 5. Let H be a Hilbert space. The linear map A : H H satisfies ( x,y H ) h Ax | y i = h
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Unformatted text preview: x | Ay i ; prove that it is automatically continuous. 6. State the Radon-Nikodym theorem (for -nite measure spaces). Show that if measures , and on ( , F ) satisfy and then d d = d d d d 7. Let 1 6 p < q < . Show that L q ( ) * L p ( ) i ( , F , ) contains measurable sets of arbitrarily large positive measure. 8. Let p,q > 1 satisfy 1 p + 1 q = 1 and let ( , F , ) be a nite measure space on which g is a measurable scalar-valued function. Prove that pointwise multiplication by g denes a bounded linear map from L p ( ) to L 1 ( ) i | g | q is integrable. 1...
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This note was uploaded on 07/08/2011 for the course MHF 3202 taught by Professor Larson during the Spring '09 term at University of Florida.

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