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Letting the (real or complex) Banach space $F$ be linearly homeomorphic to the (strong) dual of some Banach space, one sees that there is a closed topological linear subspace $E$ in $F'_\beta$ such that the map $\iota:y\mapsto{\rm ev}_y|E$ with ${\rm ev}_y|E(u)=u(y)$ becomes a linear homeomorphism $F\to E'_\beta$. Using Hahn−Banach, one easily sees that there cannot exist two different such subspaces with one a subspace of the other. However, I can't see why there might not exist two such subspaces with no one a subspace of the other. So I ask

Question. Is the space $E$ unique?

A counterexample, proof or reference will be appreciated. Note that making $E$ bigger tends to destroy surjectivity of $\iota$ whereas making $E$ smaller tends to destroy its injectivity. Related questions are here and here. I didn't quickly see whether a counterexample could be constructed from the ideas given there.

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Preduals are usually non-unique, see for example the article by Gilles Godefroy “Existence and uniqueness of isometric preduals: A survey”. books.google.com/… – Dmitri Pavlov Jan 17 2012 at 22:07
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 Perhaps the easiest examples are $C(K)$ with $K$ compact metric. All have duals isometrically isomorphic to $\ell_1$. – Bill Johnson Jan 17 2012 at 22:14
The examples Bill Johnson mentions are discussed in mathoverflow.net/questions/1380/… which the OP has linked to. I think the OP wants the different preduals to be concretely embedded as different subspaces of the dual, but if I recall correctly this is not hard to do once one has different "abstract" preduals. – Yemon Choi Jan 17 2012 at 22:30
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 I didn't click the links, which give answers to the OP's question. For the example I mentioned in my comment above, it is easy to write down the subspaces of $\ell_\infty = \ell_1^*$ corresponding to the various $C(K)$ spaces. – Bill Johnson Jan 17 2012 at 22:51
I am voting to close, for reasons somewhere between "exact duplicate" and "no longer relevant", since it seems that the question can be resolved in a straightforward way using information in the links from the OP. – Yemon Choi Jan 17 2012 at 23:37
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closed as no longer relevant by Yemon Choi, Bill Johnson, TaQ, André Henriques, Matthew Daws Jan 18 2012 at 8:49

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