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 $Y^*$)?
Functional Analysis – Is Any Banach Space a Dual Space?
banach-spacesdual-spacesfunctional-analysisgeneral-topologytopological-vector-spaces
Best Answer
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.