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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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    The validity of the Parallelogram Law is a necessary and sufficient condition for the existence of a scalar product inducing the given norm, and in the affirmative case the scalar product is uniquely determined by the Polarization identity. I have to search a reference perhaps Serge Lang Real and Functional Analysis.2012-05-26
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    Thanks t.b. . That link is exactly what I am looking for. It's a shame someone has already asked it and has got loads of points for the question (I quite like points lol). Edit: btw I voted to close. I encourage a few other people who are reading this to vote to close.2012-05-27
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    Or should I just delete this thread?2012-05-27
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    @Adam: Just leave it. There's nothing wrong with closed questions. They make finding the answers easier.2012-05-27

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