Talk:Inner product

"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, y ⟩y, where an element y ∈ X 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)