Normed spaces and the parallelogram identity

[quasar987]

Homework Statement

My professor stated the theorem "If (X,<,>) is an an inner product space and || || is the norm generated by <,>, then we have ||x+y||² + ||x-y||² = 2(||x||² + ||y||²)." But then she also said that the converse was true. I suppose this means that "Given (X, || ||) a normed space, if it satisfies the parallelogram identity, then the norm is issued from an inner product."

I do not have an idea as to how to prove that converse.

 

[morphism]

You can find this in a lot of textbooks and websites, e.g. http://www.google.ca/search?q=parallelogram+identity+inner+product. The http://www.math.uci.edu/~toikhber/TEACH/F05/M260A/NOTES/PARAL/ineq.pdf outlines the proof (or, rather, a proof) in a bunch of exercises.