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Possible Duplicate:
Norms Induced by Inner Products and the Parallelogram Law

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So suppose we are given a norm on a vector space.

If the Parallelogram law holds does that automatically mean we have the inner product which we can find using the Polarisation identity? Or is showing the Parallelogram law holds not sufficient to show that there exists an associated inner product?

Also, given that the Parallelgram law fails, e.g. $\Vert(x_1,x_2)\Vert_1 = |x_1| + |x_2|$ in $\ell^1(2)$, is there any significance in considering the Polarisation identity?

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    @Adam: Just leave it. There's nothing wrong with closed questions. They make finding the answers easier.2012-05-27

1 Answers 1

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Well, you know the definition of angle in an IPS? You can use this to find the i.p. if you know the norm and angle between two vectors: $x\cdot y=||x||\cdot||y||\cdot\cos\theta$ with $\,\theta\,=$ the angle between vectors $\,x\,,\,y\,$

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    Sure, if you are given what the angle is then your formula becomes useful. Also, doesn't your formula only work for real inner product spaces? Or is theta allowed to be complex? But anyway my point is that "they telling you what the angle is" assumes that you are being taught by someone. If you are investigating IPS on your own then there there is no one to tell your what the angle is.2012-06-05