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Do inner products come from a norm, or norms come from an inner product?

How do I prove that this is true?

Just want to know what comes from what, the chicken or the egg came first kind of thing.

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    Every inner product gives rise to a norm, but not every norm comes from an inner product. (For example, $|\cdot|_p$ in $\mathbb R^n$ corresponds to an inner product only if $n\le 1$ or $p=2$). – hmakholm left over Monica Jun 17 '16 at 16:54

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An inner product $\langle \cdot ,\cdot \rangle$ gives the norm $||x|| = \langle x ,x \rangle^{1/2}$. The axioms for a norm here are fairly easy to verify. In general, an inner product does not come from a norm. However, a norm $||\cdot||$is induced by an inner product if and only if it satisfies the parallelogram law: $$||x+y||^2 + ||x-y||^2 = 2(||x||^2+||y||^2)$$ If you want to see how to prove this, I think this answer: (Norms Induced by Inner Products and the Parallelogram Law) is very helpful. In response to "what came first", the inner product is a natural generalization of the standard dot product from Euclidean geometry. Of course the dot product also comes with an induced norm so I guess the two concepts appeared simultaneously.

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GiantTortoise1729
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