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Let $(E,\left\lVert . \right\rVert)$ be a normed vector space defined on $\mathbb{R}$ .

We suppose that the norm satisfies the Parallelogram law.

Prove that:

$$f(x,y)=(1/4)[(\left\lVert x+y \right\rVert)² - (\left\lVert x-y \right\rVert)^2]$$

defines an inner product.

I don't know how to prove the linearity of the function $f(x,y)$. In other words, how to show:

$f(kx+my,z)=k f(x,y)+m f(x,z)$.


marked as duplicate by Martin R, Clement C., hardmath, Alex M., Community Jan 18 '16 at 18:37

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  • $\begingroup$ You need to use dollar signs to properly show math formulas. $\endgroup$ – Silvia Ghinassi Jan 18 '16 at 17:56
  • $\begingroup$ As you can see in the duplicate Question, it's a good idea to treat the linearity as the combination of two results, additivity in the first argument and preservation of a scalar multiple in the first argument. $\endgroup$ – hardmath Jan 18 '16 at 18:06
  • $\begingroup$ I read it and I think I fully get it..I have a question on Step 3 (the linearity when multiplying by a scalar)..whe he proved it for the set of rational numbers and he concluded that it holds for real numbers..also i wonder if the density of the rationals in the real numbers can get us there... $\endgroup$ – Cynic Yahya Jan 25 '16 at 19:11