Talk:Inner product

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"it possible to define the geometric operation of projection onto a closed subspace" --- not quite so; rather, onto a complete subspace. It is the same in finite dimensions, and in Hilbert spaces, but not the same in incomplete (inf-dim) inner product spaces. --Boris Tsirelson 14:51, 21 June 2011 (EDT)

I changed "closed" to "complete" (I trust Boris on this). Also I introduced a field 𝔽 that is not necessarily a subfield of ℂ, and redefined the inner product as a map to 𝔽 rather than to ℂ. It seems to me that a space over a proper subfield 𝔽 ⊂ ℂ with an inner product in ℂ is hardly useful because constructions like ⟨ x, yy, where an element yX is multiplied by an inner product in ℂ, appear frequently in applications such as Fourier analysis and quantum mechanics. Such constructs would be forbidden if the space X were restricted to ground field 𝔽 that is a proper subfield of ℂ. --Paul Wormer 06:39, 1 July 2011 (EDT)
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