Is any Banach space a dual space?

general-topology functional-analysis dual-spaces banach-spaces topological-vector-spaces

Let $X$ be a Banach space. Is there always a normed vector space $Y$ such that $X$ and $Y^*$ are isometric or isomorphic as topological vector spaces (that is, there exists a linear homeomorphism between $X$ and $Y^*$)?


Solution 1:

I just sum up the answers given above to refer to my question as answered:

My favorite argument is the one given by commenter here using Krein-Milmann theorem to prove that $C_0(K)$ has no predual space.

A good reference for such questions seems to be Topics in in Banach Space Theory by Kalton and Albiac. It is used here and here to prove that $C_0(K)$ and $L_1[0,1]$ have no predual space.

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