Scalar product

[Filip84]

Hi,

can somebody help me with the problem:

Suppose that in a vector space over field of real numbers a positive defined norm is defined for each vector which satisfies the triangle inequality and ||aU||=|a|*||u||. Show that a real valued scalar product can de defined as follows:

(U,V)=1/2*(||U+V||^2 - ||U||^2- ||V||^2)

which satisfied the postulates, if following identity is satisfied by the norms:

||U+V||^2 + ||U-V||^2 = 2*||U||^2 + 2*||V||^2

 

[HallsofIvy]

All you are required to do is to show that this is a valid scalar product which means, just as the problem says, showing that it satisfies the definition of an scalar product. What is that definition?