Normed space.. inner product space

Question

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?

Answer 1

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 $x,y \in X$.

Answer 2

The norm must satisfy the parallelogram identity, i.e., you must have $||x+y|| ^2 + || x-y||^2 = 2 ( ||x||^2 + ||y||^2)$.

Look at the post here Norms Induced by Inner Products and the Parallelogram Law for a proof. (The complex case is done similarly)