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

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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 =
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Unformatted text preview: x | Ay i ; prove that it is automatically continuous. 6. State the Radon-Nikodym theorem (for σ-finite measure spaces). Show that if measures λ,μ and ν on (Ω , F ) satisfy ν ± μ and μ ± λ then d ν d λ = d ν d μ d μ d λ 7. Let 1 6 p < q < ∞ . Show that L q ( μ ) * L p ( μ ) iff (Ω , F ,μ ) contains measurable sets of arbitrarily large positive measure. 8. Let p,q > 1 satisfy 1 p + 1 q = 1 and let (Ω , F ,μ ) be a finite measure space on which g is a measurable scalar-valued function. Prove that pointwise multiplication by g defines a bounded linear map from L p ( μ ) to L 1 ( μ ) iff | g | q is integrable. 1...
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