# Talk:Inner product

From Knowino

"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 š¯”½ that is not necessarily a subfield of ā„‚, and redefined the inner product as a map to š¯”½ rather than to ā„‚. It seems to me that a space over a proper subfield š¯”½ ⊂ ā„‚ with an inner product in ā„‚ is hardly useful because constructions like ⟨
*x*,*y*⟩*y*, where an element*y*∈*X*is multiplied by an inner product in ā„‚, 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 š¯”½ that is a proper subfield of ā„‚. --Paul Wormer 06:39, 1 July 2011 (EDT)