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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.
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.
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.
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.
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.
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: $\mathcal{E}'(X) \simeq [ [X,\mathbb{R}], \mathbb{R}]_{\mathbb{R}}$?
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).
??
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?
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
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