Miscellanycont - sequences ( z n ) n = 1 such that ( 1 n (...

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Analysis 1 Miscellaneous Problems continued 1. Let X be a normed space. The series n = 1 x n in X is defined to be absolutely convergent i the real series n = 1 k x n k is convergent. Prove that if every absolutely convergent series is convergent (in the usual sense, that its sequence of partial sums converges) then X is complete. [ Suggestion : hit the gas pedal.] 2. Give X = C [0 , 1] the sup norm as usual; consider its subsets Z 1 = { f X : f (0) = 0 } , Z 2 = { f X : f (0) = f (1) = 0 } . Show that Z 1 is a closed subspace of X . To which standard space is the quotient X / Z 1 isometrically isomorphic? What about Z 2 and X / Z 2 ? 3. Let the sequence ( T n ) n = 1 L ( X , Y ) be uniformly bounded (that is, 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 . Prove further that T : Z Y : z 7→ lim n →∞ T n z is a bounded linear map. 4. Let (nonstandard notation alert!) ´ c be the set comprising all bounded scalar
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Unformatted text preview: sequences ( z n ) n = 1 such that ( 1 n ( z 1 + + z n )) n = 1 is convergent. Prove that c is a closed subspace of (relative to the sup norm). Is c separable? 5. Let Z be the space c 00 equipped with the 1 norm. Let X Z and Y Z be dened by declaring that ( z n ) n = 1 Z lies in X i z 2 n-1 = 0 for all n and in Y i z 2 n = nz 2 n-1 for all n . Prove that Z is the direct sum of its closed subspaces X and Y . Is the projection Z Y : x y 7 y a bounded linear map? 6. Prove that the normed space X is separable whenever its dual X * is separable. What about the converse? Remark : Do not be surprised (or alarmed) if one of these problems presents a challenge. 1...
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