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2011 May - X ⊥ Y then X Y is closed 6 Let(Ω F,μ be a...

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Analysis PhD Examination May 2011 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. (i) State the Hahn-Banach theorem. (ii) Show that there exists an isometric linear map from each separable normed space into ‘ ∞ . 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. State the Closed Graph Theorem and the Banach Isomorphism Theorem; deduce one of these from the other. 4. Let the sequence ( T n ) ∞ n =1 ⊂ L ( X, Y ) be bounded in operator norm and assume Y complete. Prove that Z = { z ∈ X : ( T n z ) ∞ n =1 converges } is a closed subspace of X . 5. Let X and Y are closed subspaces of a Hilbert space. Prove that if
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Unformatted text preview: X ⊥ Y then X + Y is closed. 6. Let (Ω , F ,μ ) be a measure space. Define F to comprise all those A ⊆ Ω for which there exist L,U ∈ F such that L ⊆ A ⊆ U and μ ( U r L ) = 0 and then define μ ( A ) to be the common value μ ( L ) = μ ( U ). Show that (Ω , F , μ ) is a measure space that is complete in the sense that each subset of a null set is null. 7. Let (Ω , F ,μ ) be an arbitrary measure space; let p,q,r > 1 satisfy 1 p + 1 q = 1 r . Prove that if f ∈ L p ( μ ) and g ∈ L q ( μ ) then fg ∈ L r ( μ ) with k fg k r 6 k f k p k g k q . 8. Let 1 6 p < q < ∞ . Show that L p ( μ ) * L q ( μ ) iff (Ω , F ,μ ) contains measurable sets of arbitrarily small positive measure. 1...
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