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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 13th 2017
   • (edited Aug 13th 2017)
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   Author: Urs
   Format: MarkdownItexI have started to add some of the basic definitions and facts to _[[Schwartz space]]_, _[[tempered distribution]]_ and _[[Fourier transform of distributions]]_. Notice that we had an entry titled "Schwartz space" already since May 2013 ([rev 1](https://ncatlab.org/nlab/revision/Schwartz+space/1) by Andrew Stacey) which considered not spaces of smooth functions with rapidly decreasing derivatives, but locally convex TVSs $E$ "with the property that whenever $U$ is an [[absolutely convex]] neighbourhood of $0$ then it contains another, say $V$, such that $U$ maps to a [[precompact set]] in the [[normed vector space]] $E_V$." I had not been aware of this use of "Schwartz space" before, and Andrew gave neither reference nor discussion of the evident question, whether "the" Schwartz space is "a" Schwartz space. In June 2015 somebody saw our entry and shared his confusion about this point on Maths.SE [here](https://math.stackexchange.com/q/1317874/58526), with no reply so far. I see that this other use of "Schwartz space" appears in Terzioglu 69 ([web](https://link.springer.com/article/10.1007/BF01350326)) where it is attributed to Grothendieck.

   I have started to add some of the basic definitions and facts to Schwartz space, tempered distribution and Fourier transform of distributions.

   Notice that we had an entry titled “Schwartz space” already since May 2013 (rev 1 by Andrew Stacey) which considered not spaces of smooth functions with rapidly decreasing derivatives, but locally convex TVSs $E$ “with the property that whenever $U$ is an absolutely convex neighbourhood of $0$ then it contains another, say $V$, such that $U$ maps to a precompact set in the normed vector space $E_V$.”

   I had not been aware of this use of “Schwartz space” before, and Andrew gave neither reference nor discussion of the evident question, whether “the” Schwartz space is “a” Schwartz space. In June 2015 somebody saw our entry and shared his confusion about this point on Maths.SE here, with no reply so far.

   I see that this other use of “Schwartz space” appears in Terzioglu 69 (web) where it is attributed to Grothendieck.

2. • CommentRowNumber2.
   • CommentAuthorAndrew Stacey
   • CommentTimeAug 14th 2017
   • (edited Aug 14th 2017)
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   Author: Andrew Stacey
   Format: MarkdownItex(Just driving by ...) I got the definition from (where else?) [The Convenient Setting of Global Analysis](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf), where it is given on p585 after Result 52.24 (that result is referenced from _Jarchow, 1981, 10.4.3, p202_ and _Horvath, 1966, p277_). It wouldn't surprise me that it went back to Grothendieck, nor even that I knew of that at the time. I just had a quick look in Pietsch's book on Nuclear Spaces, there was stuff on s-Nuclear spaces but not on Schwartz spaces - however, I suspect that the two may be the same. Certainly _the_ Schwartz space is _a_ Schwartz space. The reason for this definition is to figure out what characterises _the_ Schwartz space in terms of functional analytic properties. Oh, and I replied to the Math.SE question for the sake of completeness.

   (Just driving by …)

   I got the definition from (where else?) The Convenient Setting of Global Analysis, where it is given on p585 after Result 52.24 (that result is referenced from Jarchow, 1981, 10.4.3, p202 and Horvath, 1966, p277). It wouldn’t surprise me that it went back to Grothendieck, nor even that I knew of that at the time. I just had a quick look in Pietsch’s book on Nuclear Spaces, there was stuff on s-Nuclear spaces but not on Schwartz spaces - however, I suspect that the two may be the same.

   Certainly the Schwartz space is a Schwartz space. The reason for this definition is to figure out what characterises the Schwartz space in terms of functional analytic properties.

   Oh, and I replied to the Math.SE question for the sake of completeness.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 6th 2017
   • (edited Sep 6th 2017)
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   Author: Urs
   Format: MarkdownItex(belated reply, had missed this while being on vacation) Thanks, Andrew! I have edited the entry _[[Schwartz space]]_ a little, following your pointers.

   (belated reply, had missed this while being on vacation)

   Thanks, Andrew! I have edited the entry Schwartz space a little, following your pointers.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 7th 2017
   • (edited Sep 7th 2017)
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   Author: Urs
   Format: MarkdownItexand I have cross-linked with _[[Fréchet space]]_: [here](https://ncatlab.org/nlab/show/Fr%C3%A9chet+space#SchwartSpaceOfFunctionsIsFrechetSpace)

   and I have cross-linked with Fréchet space: here