Talk:Parallelogram law

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For normed vector spaces whose norm obeys the parallelogram law, the operation

\langle x, y\rangle={\|x+y\|^2-\|x\|^2-\|y\|^2\over 2}

is an inner product only for a real vector space, i.e. a vector space over some scalar field that is contained within the reals. For the general case of complex vector spaces we need to define

\langle x | y\rangle={\|x+y\|^2-\|x\|^2-\|y\|^2\over 2}+{\|x+jy\|^2-\|x\|^2-\|jy\|^2\over 2j}

where j is some nonzero pure imaginary element of the scalar field, but not necessarily \sqrt{-1} (in order to take into account fields such as \mathbb{Q}(\sqrt{-2}) that do not contain \sqrt{-1}.)

This isn't a problem when talking about complete spaces, and by the way something about the relationship between Hilbert spaces and Banach spaces could at least be mentioned in passing here as well.

130.94.162.64 23:02, 29 October 2005 (UTC)

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