Equivalent norms and inner product It is not hard to give examples of normed spaces which are not inner product spaces. Now let $(V, \|\cdot\|)$ be a normed space.
Is it always possible to construct an inner product on $V$ which gives the same topology on $V$ as before? (As pointed out in comments, in finite dimensional case it it always the case).

An inner product space (i.e. the topological vector space induced by an inner product) is reflexive (even: $X$ and $X^\ast$ are isometric), at least if $X$ is complete. And e.g. $\ell_1$ does not have this property.
So at least you need to demand that $X$ and its dual are isometric (in the complete case). And even then there probably are counterexamples.
