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 deﬁned by declaring that ( z n ) ∞ n = 1 ∈ Z lies in X i ﬀ z 2 n1 = 0 for all n and in Y i ﬀ z 2 n = nz 2 n1 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
 LARSON

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