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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?

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  • $\begingroup$ It must satisfy the parallelogram law. $\endgroup$ – Augustin May 8 '16 at 11:52
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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)

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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 $x,y \in X$.

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