# 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).

• All norms of $\mathbb{R}^n$ are equivalent. And only a few (relatively speaking) are induced by inner products. Dec 10, 2014 at 18:45
• The second question is true for $\mathbb{R}^n$, i.e. all finite-dimensional cases. But is fails for infinite-dimensional ones. Dec 10, 2014 at 18:47
• @HennoBrandsma Thanks for the comments. I have edited the question. Do you have a counterexample for the infinite dimensional setting? Dec 10, 2014 at 18:50

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.