How did Von Neumann think of the formula for scalar product?

Asked Jan 25 '16 at 03:45

Active Jan 25 '16 at 07:10

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My question is on the idea behind Von Neumann's formula for the scalar product induced from a norm that satisfies the parallelogram law. $\langle x,y\rangle=\frac 14(\|x+y\|^2-\|x-y\|^2)$. Was it by trial and error or is there some intuitive way that one can guess this formula?

I can prove that this is indeed a scalar product, it is the idea behind it that baffles me.

edited Jan 25 '16 at 07:10

Martin Sleziak

53,687

asked Jan 25 '16 at 03:45

yoyostein

19,608

That is classical elementary plane geometry. I suggest you read the Wikipedia page on the Parallelogram Law. – Mariano Suárez-Álvarez Jan 25 '16 at 03:58
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This is a well known identity for the 2-norm -- for example, this identity is used to prove that a linear transformation that preserves the 2-norm also preserves inner products. You can prove the identity easily (for the 2-norm) by expanding the right hand side. – littleO Jan 25 '16 at 05:45
@littleO thanks I get it now – yoyostein Jan 25 '16 at 05:59
I will add link to the post where the proofs about this can be found: Norms Induced by Inner Products and the Parallelogram Law. (It is not what you ask about, but if somebody stumbles upon your post, this link might be useful for them.) – Martin Sleziak Jan 25 '16 at 07:12

How did Von Neumann think of the formula for scalar product?

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