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?



Subspaces of duals
$Y=L\_1[0,1]$ 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. 


