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Unformatted text preview: Exercises from Appendix C C.10 Hint: Expand bardbl f + g bardbl 2 = bardbl Lf + Lg bardbl 2 and bardbl f + ig bardbl 2 = bardbl Lf + iLg bardbl 2 using the Polar Identity. C.14 Hint: (b) If / , then there exists a subsequence ( n k ) k N such that  n k  k for each k . Let c n k = 1 /k and define all other c n to be zero. Show that f = c n e n converges but Lf = n c n e n does not converge. C.15 Hints: (a) For the case p < , to show the inequality bardbl M bardbl bardbl bardbl , choose any > 0. Then E = {  > bardbl bardbl } has positive measure, so some set E k = E [ k,k + 1] must have positive (and finite) measure. Consider f =  E k  1 /p E k . (b) Suppose p < , and assume / L ( R ). The sets E k = { k   < k + 1 } . are measurable and disjoint, and since is not in L ( X ) there must be infinitely many E k with positive measure, say E n k for k N . Choose F k E n k with 0 <  F k  < , and consider f ( x ) = n k  F k  1 /p for x F k , and f ( x ) = 0 otherwise. C.26 Hints: Fix f X. Since Y is dense in X, there exist g n Y such that g n f . Show that { Lg n } n N is Cauchy in Z , so there exists an h Z such that Lg n h . Show that tildewide Lf = h is welldefined and has the required properties. C.24 Hints: (b) Let { A n } n N be a sequence in B ( X,Y ) that is Cauchy in operator norm. Given f X, show that { A n f } n N is a Cauchy sequence in Y and hence converges, say to g . Define Af = g . Use the fact that A n f Af for each f together with the fact that { A n } n N is Cauchy to show that A n A in operator norm. C.31 Hint: (b) Consider a function g C ( R ) that is nonzero everywhere. C.34 Hints: (b) If negationslash = 0, choose g / ker( ). Let p be the orthogonal projection of g onto ker( ), and set e = g p . Show that ( e ) negationslash = 0, and set u = e/ ( e )....
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This note was uploaded on 08/25/2011 for the course MATH 7338 taught by Professor Heil during the Fall '09 term at Georgia Institute of Technology.
 Fall '09
 HEIL

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