Talk:Parallelogram law

Latest comment: 11 years ago by Sniffnoy in topic Normed vector spaces

Normed vector spaces

edit

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

 

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

 

where   is some nonzero pure imaginary element of the scalar field, but not necessarily   (in order to take into account fields such as   that do not contain  )

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)Reply

Certainly the omission of the converse to the statement that a norm coming from an inner product satisfies the parallelogram law hold needs to be addressed, in my opinion. 143.210.42.231 (talk) 09:51, 15 May 2013 (UTC)Reply

That isn't omitted -- the section "The parallelogram law in inner product spaces" covers that. If you think it isn't clear enough, edit it. Sniffnoy (talk) 18:22, 15 May 2013 (UTC)Reply

equivalence of p.-law and general quadrilateral being a parallologram

edit

From the formula for the general quadrilateral it follows easily that a general quadrilateral is a parallelogramm if x=0. This and the formula itself are easy enough to proof, but I cannot find a quotable source for either statement. Anybody else got one?--CWitte (talk) 15:56, 12 October 2010 (UTC)Reply