Unformatted text preview: ( m , n ) → ∞ . Show that there exists a realvalued Function f in L 2 (µ) For which k f n − f k 2 → 0 as n → ∞ , by the Following steps. (i) Show that sup n ∈ N k f n k 2 < ∞ . (ii) Show that there is no loss oF generality in supposing f n ≥ 0 For all n . Hint: Consider f ± n . (iii) Show that { f 2 n : n ∈ N } is a Cauchy sequence in L 1 (µ) . (iv) Show that there exists a Function g in L 1 (µ) with g ≥ 0 and µ  f 2 n − g  → 0. (v) Show that √ g ∈ L 2 (µ) and k f n − √ g k 2 → 0. Hint:  f n − √ g  ≤ f n + √ g . (vi) Anything more to do?...
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 Spring '09
 DavidPollard
 Statistics, Probability, Continuous function, µ, measure, UGMTP Problem

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