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Please help me prove this. I'm not sure how to apply the parallelogram law to the norm.

  • 1
    $\begingroup$ What norm?${}{}{}{}{}$ $\endgroup$ – Omnomnomnom May 17 '15 at 2:21
  • $\begingroup$ I notice this small paragraph uses inconsistent terminology: "IP" and "inner product" and "scalar product," it is not clear if these are all the same thing. I think the parallelogram law stated is under the assumption that you have an inner product $<x,y>$ and you define $||x||^2 = <x,x>$. $\endgroup$ – Michael May 17 '15 at 2:23
  • $\begingroup$ I guess it's the $\ell^1$ norm mentioned in the page. @OP: just show the norm does not satisfy the Parallelogram Law. $\endgroup$ – Quang Hoang May 17 '15 at 2:24
  • $\begingroup$ I think it just wants me to prove that the norm does not come from an inner product. I know I have to show it does not satisfy the parallelogram law but I don't know how to apply this. $\endgroup$ – user241173 May 17 '15 at 2:49

Pick two vectors $x$, $y$ in the space under consideration, which is presumably $\mathbb{R}^2$.

Calculate $\|x+y\|$, $\|x-y\|$, $\|x\|$ and $\|y\|$ using the norm you are given, in this case $\|\cdot\|_1$.

See whether the parallelogram rule holds for this particular $x$, $y$.

If it fails, then you know that the norm does not come from an inner product.


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