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?

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- Oct 7th 2011, 12:38 PMCamille91Bilinear of dot product
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?

- Oct 7th 2011, 01:08 PMOpalgRe: Bilinear of dot product
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 , but I doubt it.