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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 13th 2009
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownI wrote the article [[distribution]]. I'm by no means an expert though. I left open a section "Applications" in case someone would like to add some, or if not I'll try to fill this in soon.

   I wrote the article distribution. I'm by no means an expert though. I left open a section "Applications" in case someone would like to add some, or if not I'll try to fill this in soon.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 13th 2009
   • PermaLink
   Author: Urs
   Format: MarkdownI added to [[distribution]] a section on their formulation in [[synthetic differential geometry]]. So far I just say a word and then give the reference. But it would be nice to further expand on this, eventually. I also added an automatic table of contents.

   I added to distribution a section on their formulation in synthetic differential geometry.

   So far I just say a word and then give the reference. But it would be nice to further expand on this, eventually.

   I also added an automatic table of contents.

3. • CommentRowNumber3.
   • CommentAuthorAndrew Stacey
   • CommentTimeOct 14th 2009
   • PermaLink
   Author: Andrew Stacey
   Format: MarkdownI modified the part on operators. One usually is not so interested in _dual_ operators but _extension_ operators. This explains, by the way, the sign in the formula for the derivative of a distribution. (I taught Fourier analysis last year and got myself horrendously confused until I straightened this out in my head!) I've also mentioned briefly that there are other topologies than the weak* topology. Indeed, one could argue that the strong topology is much more natural than the weak* topology.

   I modified the part on operators. One usually is not so interested in dual operators but extension operators. This explains, by the way, the sign in the formula for the derivative of a distribution.

   (I taught Fourier analysis last year and got myself horrendously confused until I straightened this out in my head!)

   I've also mentioned briefly that there are other topologies than the weak* topology. Indeed, one could argue that the strong topology is much more natural than the weak* topology.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 7th 2017
   • (edited Sep 7th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added pointer ([here](https://ncatlab.org/nlab/show/distribution#Paugam12)) to Frédéric Paugam's discussion of distributions as morphisms of "smooth sets" (sheaves on CartSp) in section 3.2 of his book ([pdf](https://webusers.imj-prg.fr/~frederic.paugam/documents/enseignement/master-mathematical-physics.pdf)). In particular his observation in prop. 3.2.11 says, if I am understanding well, that compactly supported distributions on $\mathbb{R}^n$ are precisely just the $\mathbb{R}$-linear morphisms from the internal hom $[\mathbb{R}^n, \mathbb{R}]$ to $\mathbb{R}$ in $SmoothSet \coloneqq Sh(CartSp) \hookrightarrow Smooth\infty Grpd$. That must be a useful perspective. I need to come back to this later when I am more awake.

   I have added pointer (here) to Frédéric Paugam’s discussion of distributions as morphisms of “smooth sets” (sheaves on CartSp) in section 3.2 of his book (pdf).

   In particular his observation in prop. 3.2.11 says, if I am understanding well, that compactly supported distributions on $\mathbb{R}^n$ are precisely just the $\mathbb{R}$-linear morphisms from the internal hom $[\mathbb{R}^n, \mathbb{R}]$ to $\mathbb{R}$ in $SmoothSet \coloneqq Sh(CartSp) \hookrightarrow Smooth\infty Grpd$.

   That must be a useful perspective. I need to come back to this later when I am more awake.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 7th 2017
   • (edited Sep 7th 2017)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSo that's a version of what's at [[compactly supported distribution]]? >the compactly supported distributions exhaust the continuous linear functionals on the space of smooth functions I guess it's pointing in a more general abstract direction.

   So that’s a version of what’s at compactly supported distribution?

   the compactly supported distributions exhaust the continuous linear functionals on the space of smooth functions

   I guess it’s pointing in a more general abstract direction.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeSep 7th 2017
   • (edited Sep 7th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, but the claim is that instead of talking about continuity of functionals with respect to topologies on function spaces laboriously established, we may simply form the maps in the sheaf topos out of the internal hom, i.e. simply consider _linear diffeological maps_ (or their Cahiers topos refinement). The bijective identification of distributions with such linear diffeological maps is not restricted to compactly supported distributions. I see now that the general statement that $$ \mathcal{D}'(X) \simeq Hom_{Topos}(\mathcal{D}(X), \mathbb{R})_{\mathbb{R}} \phantom{AAA} \text{as sets} $$ is also in [Kock-Reyes 04, prop. 72](https://ncatlab.org/nlab/show/distribution#KockReyes04) for the Cahiers topos, as well as prop. 3.2.11 in Frédéric Paugam’s book ([pdf](https://webusers.imj-prg.fr/~frederic.paugam/documents/enseignement/master-mathematical-physics.pdf)) for smooth sets (sheaves on $CartSp$). What I don't understand yet is what the claim is regarding the internal linear mapping space $[\mathcal{D}(X), \mathbb{R}]_{\mathbb{R}}$ etc. Maybe that's not equivalent to the image of the convenient vector space $\mathcal{D}'(X)$ in the topos? Maybe that only becomes equivalent if we consider compactly supported distributions: $\mathcal{E}'(X) \simeq [ [X,\mathbb{R}], \mathbb{R}]_{\mathbb{R}}$?

   Yes, but the claim is that instead of talking about continuity of functionals with respect to topologies on function spaces laboriously established, we may simply form the maps in the sheaf topos out of the internal hom, i.e. simply consider linear diffeological maps (or their Cahiers topos refinement).

   The bijective identification of distributions with such linear diffeological maps is not restricted to compactly supported distributions. I see now that the general statement that

   $$\mathcal{D}'(X) \simeq Hom_{Topos}(\mathcal{D}(X), \mathbb{R})_{\mathbb{R}} \phantom{AAA} \text{as sets}$$

   is also in Kock-Reyes 04, prop. 72 for the Cahiers topos, as well as prop. 3.2.11 in Frédéric Paugam’s book (pdf) for smooth sets (sheaves on $CartSp$).

   What I don’t understand yet is what the claim is regarding the internal linear mapping space $[\mathcal{D}(X), \mathbb{R}]_{\mathbb{R}}$ etc. Maybe that’s not equivalent to the image of the convenient vector space $\mathcal{D}'(X)$ in the topos? Maybe that only becomes equivalent if we consider compactly supported distributions: $\mathcal{E}'(X) \simeq [ [X,\mathbb{R}], \mathbb{R}]_{\mathbb{R}}$?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeSep 7th 2017
   • (edited Sep 7th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am feeling a bit stuck. Maybe it's too late at night. I am stuck already on the simplest case: For $X$ a smooth manifold, there is on $C^\infty(X)$ the topology of uniform convergence over all compact subspaces for all derivatives. A compactly-supported distribution is a linear functional $$ C^\infty(X) \longrightarrow \mathbb{R} $$ continuous with respect to this topology. Why is this equivalently an $\mathbb{R}$-linear morphism $$ [X,\mathbb{R}] \longrightarrow \mathbb{R} $$ a) of [[smooth sets]] (sheaves over $CartSp$ as in the reference in #4)? and/or b) of objects in the Cahiers topos (as in the reference in #6). ??

   I am feeling a bit stuck. Maybe it’s too late at night. I am stuck already on the simplest case:

   For $X$ a smooth manifold, there is on $C^\infty(X)$ the topology of uniform convergence over all compact subspaces for all derivatives. A compactly-supported distribution is a linear functional

   $$C^\infty(X) \longrightarrow \mathbb{R}$$

   continuous with respect to this topology.

   Why is this equivalently an $\mathbb{R}$-linear morphism

   $$[X,\mathbb{R}] \longrightarrow \mathbb{R}$$

   a) of smooth sets (sheaves over $CartSp$ as in the reference in #4)?

   and/or

   b) of objects in the Cahiers topos (as in the reference in #6).

   ??

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 8th 2017
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexSo Paugam by chapter 10 has turned to the microlocalisation approach, subanalytic sheaves and so on (e.g., 10.4.2). Is there supposed to be a connection with the material on distributions presented in sec 3.2?

   So Paugam by chapter 10 has turned to the microlocalisation approach, subanalytic sheaves and so on (e.g., 10.4.2). Is there supposed to be a connection with the material on distributions presented in sec 3.2?

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 10th 2017
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   Author: DavidRoberts
   Format: MarkdownItexIn the absence of a more obvious place to mention this, the notes for the course [Analysis on manifolds](https://www.staff.science.uu.nl/~ban00101/anman2009/anman2009.html) are very good and take a modern, axiomatic viewpoint to various spaces of distributions and functions that might be useful (eg lecture 3 is quite good). They aren't mentioned at [[distribution]] but may have been already added elsewhere. I haven't time to hunt down all the places where they might be added, so I will just put them at [[distribution#traditional]]

   In the absence of a more obvious place to mention this, the notes for the course Analysis on manifolds are very good and take a modern, axiomatic viewpoint to various spaces of distributions and functions that might be useful (eg lecture 3 is quite good). They aren’t mentioned at distribution but may have been already added elsewhere. I haven’t time to hunt down all the places where they might be added, so I will just put them at distribution#traditional

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2017
    • (edited Oct 18th 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexRegarding #7: as I have announced already in another thread, I am now giving this discussion its dedicted entry at _[[distributions are the smooth linear functionals]]_. Accorddingly I have renamed the two subsections at _[[distribution]]_ which used to be called 1. Definition -- Traditional 1. Definition -- In terms of monads and had zero content under "In terms of monads" into 1. Definition -- As continuous linear functionals 1. Definition -- As smooth linear functionals and then I gave the second of these a little bit of Idea-text and then a pointer to that new entry _[[distributions are the smooth linear functionals]]_.

    Regarding #7: as I have announced already in another thread, I am now giving this discussion its dedicted entry at distributions are the smooth linear functionals.

    Accorddingly I have renamed the two subsections at distribution which used to be called

    1. Definition – Traditional

    2. Definition – In terms of monads

    and had zero content under “In terms of monads”

    into

    1. Definition – As continuous linear functionals

    2. Definition – As smooth linear functionals

    and then I gave the second of these a little bit of Idea-text and then a pointer to that new entry distributions are the smooth linear functionals.