TITLE: Is every normed vector space, an inner product space
QUESTION [15 upvotes]: Let $V$ be a vector space over  $\mathbb{C}$. If $V$ is an inner product space, then $V$  is normed (where the norm is defined as $\|x\|=\sqrt{(x,x)}\,\,$). Now if $V$ is normed, does it follow that $V$ is an inner product space ? I suspect no. I would like to see an example.
Thank you.

After reading my question again, I think it needs some clarification:
Suppose that $V$ is normed with norm $||\,||$. Can $V$ be given an inner product space structure such that $(x,x)=||x||^2$ ?

REPLY [8 votes]: Inner product spaces satisfy the parallelogram law.
If you can find a counterexample to the parallegram law in a space with the sup norm, then you've found a normed linear space that is not an inner product space.
Even more interestingly, inner product spaces are the ONLY normed vector spaces that satisfy the parallelogram law.