# Normed space.. inner product space

Every inner product space is normed space but the converse is not true, what is the condition we must have to make the converse true?

• It must satisfy the parallelogram law. – Augustin May 8 '16 at 11:52

The norm must satisfy the parallelogram identity, i.e., you must have $||x+y|| ^2 + || x-y||^2 = 2 ( ||x||^2 + ||y||^2)$.
Let $(X,\|\cdot\|)$ be a normed vector space. Then, $\|\cdot\|$ arises from an inner product if, and only if, it satisfies the parallelogram law:
$$\|x+y\|^2 + \|x - y\|^2 = 2(\|x\|^2 + \|y\|^2)$$
for all $x,y \in X$.