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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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- Spring '09