(Lax, Ch. 8, Exercise 3) Show that if is a closed subspace of a normed linear space , then is isometrically isomorphic to the quotient of by the annihilator of in . (The quotient is given the quotient topology.) At some point you need Hahn-Banach.
Fill in the details of the following point which was mentioned in class but skipped over quickly: If is a Banach space and is the canonical map , show that is closed in in the norm topology. (Hint: The completeness of is essential here, along with the fact that is an isometry.)
Show that if a Banach space is isometric to the dual of another Banach space , then there is a norm-one projection from onto . (Hint: consider , where is the canonical injection of into .)
Prove that the Banach space of sequences tending to 0 is not the dual space of any other Banach space. Here's a sketch. Assume that is the dual of a Banach space . Then (by #2), there is a linear projection with and with norm 1. Let and suppose you can show there is an infinite subset of such that if is supported in , then . Then , i.e., every supported in tends to 0, which is clearly false. To construct , first show (this is just combinatorics) that there are uncountably many infinite subsets of ( running over an uncountable index set ) with the property that any two of these have finite intersection. Suppose you could find supported in for each with . (Otherwise there is some for which if is supported in .) Then use the property of the and take linear combinations of the to contradict the fact that has norm 1.
Can't change a rubric once you've started using it.