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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 7th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have written an Idea-section at _[[operator-valued distribution]]_.

   I have written an Idea-section at operator-valued distribution.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeAug 9th 2017
   • (edited Aug 9th 2017)
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   Author: zskoda
   Format: MarkdownItexBerezansky/Berezanskii has written several books on infinite-dimensional analysis, e.g., * Yu.M. Berezansky, Y.G. Kondratiev, _Spectral methods in infinite-dimensional analysis_, [Springerlink](http://www.springer.com/us/book/9780792328490) 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](http://dic.academic.ru/dic.nsf/enc_mathematics/3760/%D0%9E%D0%A1%D0%9D%D0%90%D0%A9%D0%95%D0%9D%D0%9D%D0%9E%D0%95)) 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.

   Berezansky/Berezanskii has written several books on infinite-dimensional analysis, e.g.,

   • Yu.M. Berezansky, Y.G. Kondratiev, Spectral methods in infinite-dimensional analysis, Springerlink

   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.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeAug 9th 2017
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI 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 * N. N. Bogoliubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, _General principles of quantum field theory_, Kluwer 1990 (Ch. 2 sec. 7)

   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

   • N. N. Bogoliubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, General principles of quantum field theory, Kluwer 1990 (Ch. 2 sec. 7)
4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeAug 9th 2017
   • (edited Aug 9th 2017)
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   Author: Urs
   Format: MarkdownItexThanks 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](https://ncatlab.org/nlab/show/operator-valued+distribution#Definition). 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.)

   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.)

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeAug 9th 2017
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThanx for excellent comments and corrections.

   Thanx for excellent comments and corrections.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeSep 7th 2017
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI 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.

   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.