Thread: Bilinear of dot product

How to prove that if for $\displaystyle ( \mathbb{R}^n , \ || \ * \ ||) $ is $\displaystyle ||x+y||^2+||x-y||^2=2(||x||^2+||y||^2) $ then we can define a dot product as $\displaystyle <x,y>=\frac{||x+y||^2-||x-y||^2}{4}$? Intuitionly it is true but I have great problem with proof of bilinear of it. It doesn't seem so simple. Any clues?

Originally Posted by Camille91

How to prove that if for $\displaystyle ( \mathbb{R}^n , \ || \ * \ ||) $ is $\displaystyle ||x+y||^2+||x-y||^2=2(||x||^2+||y||^2) $ then we can define a dot product as $\displaystyle <x,y>=\frac{||x+y||^2-||x-y||^2}{4}$? Intuitionly it is true but I have great problem with proof of bilinear of it. It doesn't seem so simple. Any clues?

This is the Jordan–von Neumann theorem, and it certainly isn't simple. Unless you are another von Neumann, you are likely to need more than a few clues to find a proof of it. You can find a proof here. (The proof is given for a complex space, but there is a footnote indicating that the proof for a real space is similar.)

Edit. The J–vN theorem is actually for infinite-dimensional spaces. I suppose there might conceivably be a simpler proof for the finite-dimensional space $\displaystyle \mathbb{R}^n$, but I doubt it.