Unformatted text preview: x  Ay i ; prove that it is automatically continuous. 6. State the RadonNikodym theorem (for σﬁnite 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 scalarvalued 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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 Spring '09
 LARSON
 Hilbert space, Topological space, PhD Examination

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