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| ``Re: parens''
by lars_h on 2005-07-01 05:04:43 |
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| > 1. Physicists generally use angle brackets with a vertical
> bar instead of the comma: <.|.>. This is the inner product
> in Dirac's bra-ket notation.
Yes, although that is because they use |.> and <.| in separation,
the former usually for elements of a vector space and the latter
for functionals on that vector space, so they "really" leave one
of the bars out when they apply functional to vector.
> 2. From the footnote: "A small minority of authors impose
> linearity on the second coordinate instead of the first
> coordinate." Perhaps, if "authors" is replaced with
> "mathematicians". Essentially all, if not all, physics
> texts impose linearity on the second coordinate.
Since this footnote appears in the entry for something as basic
as "inner product", the linearity business really has to be
explained more fully. It's only an issue for complex vector
spaces, so maybe the definition is best stated first for real
vector spaces (noting that the inner product is linear in both
factors), and then again for complex vector spaces (with a remark
that it has to be linear in one factor and conjugate linear (?)
in the other, noting various conventions on the matter).
What's more, I have seen definitions of inner products also
over fields of positive characteristic, so the claim that inner
products exist only over the reals and complex numbers is
disputable. |
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