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appendixc_sol

# appendixc_sol - Exercises from Appendix C C.10 Hint Expand...

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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 λ / , then there exists a subsequence ( λ n k ) k N such that | λ 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 = λ n c n e n does not converge. C.15 Hints: (a) For the case p < , to show the inequality bardbl M φ bardbl ≤ bardbl φ bardbl , choose any ε > 0. Then E = {| φ | > bardbl φ bardbl ε } has positive measure, so some set E k = E [ k, k + 1] must have positive (and finite) measure. Consider f = | E k | 1 /p χ E k . (b) Suppose p < , and assume φ / L ( R ). The sets E k = { k ≤ | φ | < k + 1 } . are measurable and disjoint, and since φ 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 well-defined 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 0 ( R ) that is nonzero everywhere. C.34 Hints: (b) If μ negationslash = 0, choose g / ker( μ ). Let p be the orthogonal projection of g onto ker( μ ), and set e = g p . Show that μ ( e ) negationslash = 0, and set u = e/μ ( e ). Show that f μ ( f ) u ker( μ ) for every f H . Use the fact that u ker( μ ) to show that μ = μ h where h = u/ bardbl u bardbl 2 .

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appendixc_sol - Exercises from Appendix C C.10 Hint Expand...

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