# Finding the inner product given the norm [duplicate]

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

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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## marked as duplicate by t.b., Adam Rubinson, Jonas Teuwen, Martin Sleziak, Rudy the ReindeerMay 27 '12 at 8:48

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. – Giuseppe Tortorella May 26 '12 at 17:10
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. – Adam Rubinson May 27 '12 at 7:34
Or should I just delete this thread? – Adam Rubinson May 27 '12 at 8:46
@Adam: Just leave it. There's nothing wrong with closed questions. They make finding the answers easier. – t.b. May 27 '12 at 9:30

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 $\,x\,,\,y\,$