Subspaces of duals It is an easy undergraduate exercise to show that (finite) direct sums are preserved under dualisation. Thus, it is natural to ask if we the following holds:
is it true that if $X$ is a subspace of $Y$, then $X^* $ is a subspace of $Y^*$? 
In many cases this is certainly not true (one can construct relevant subspaces of $Y=\ell^\infty$) but... let me say that $Y$ has property (D) if the above hypothesis holds for $Y$. 
Is it true that the only spaces with property (D) are Hilbert spaces?
 A: $Y=L_1[0,1]$ has the property (D) since it is separable and the dual of any separable space embeds into $Y^\ast = L_\infty[0,1]$.
Of course, any separable space with a complemented subspace whose dual is isomorphic to $L_\infty[0,1]$ will have the property too.
If I think of other examples that are fundamentally different I'll add them later. 
A: There are separable reflexive examples also, such as the $\ell_2$ sum $X$ of a sequence of finite dimensional spaces that is dense (in the sense of the Banach-Mazur distance) in the collection of all finite dimensional spaces. See my 1974 paper with Zippin in the Israel Journal of Mathematics.
Another example is a separable reflexive space $X$ s.t. every subspace of every quotient of $X$ is isomorphic to a complemented subspace of $X$, yet $X$ is not isomorphic to a Hilbert space. This is in a recent paper with Szankowski and can be downloaded from 
http://www.math.tamu.edu/~bill.johnson/selpubs.html
(no. 117). 
The example with Zippin lacks this property since it has subspaces that fail the approximation property (we proved in an earlier paper that each of its subspaces that has the approximation property is isomorphic to a complemented subspace).
