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I have written an Idea-section at operator-valued distribution.
Berezansky/Berezanskii has written several books on infinite-dimensional analysis, e.g.,
where not only operatros are considered but also that the base space is infinitedimensional and has had in mind applications in QFT. Some of his works are translated to English. As far as generalized functions he mainly used the Gel’fand triple approach. Now I see that we have two pages with different terminology and references which should be merged in my opinion: Gelfand triple and rigged Hilbert space. In Russian the term is оснащенное гильбертово пространство (see e.g. here) from the very beginning and it means “equipped/enriched Hilbert space”. Gelfand triple is western term. Later John Robert’s term rigged Hilbert space is a literally bit different from Russian term, but he might have freely translated.
I added the definition at operator-valued distribution.
Let $V$ be a topological vector space and $M$ a smooth manifold. A $V$-valued distribution on $M$ is a continuous linear function from the Schwarz space of test functions $\mathcal{S}(M)$ to $V$.
Sometimes of course we use different spaces of test functions.
and yet another comprehensive reference which explains it
Thanks for the additions!
Sometimes of course we use different spaces of test functions.
I suggest we consistently call them by their names:
$\mathcal{D}'$ is distributions
$\mathcal{E}'$ is compactly supported distributions
$\mathcal{S}'$ is tempered distributions.
I have slightly edited accordingly your edit here.
But the entry on operator-valued distributions is not really the place to sort this out. This should be explained properly at distribution.
Unfortunately that entry is a bit of a mess presently. Maybe one day somebody finds the energy to clean it up.
(These variants need to be cleaned up, also the issue about densities needs to be cleaned up, with that out of the way then proper discussion not just for smooth manifolds but for (pseudo-)Riemannian smooth manifolds needs to be added.)
Thanx for excellent comments and corrections.
I forgot that some time ago I wrote an entry Connes distribution which is also a variant of the idea of distributions which can be applied in the infinite-dimensional context.
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