How to prove that if for is then we can define a dot product as ? Intuitionly it is true but I have great problem with proof of bilinear of it. It doesn't seem so simple. Any clues?
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 , but I doubt it.