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This is a reference request, coming from someone with little knowledge of hyperfunctions:

Which methods have been used to endow the space of hyperfunctions $\mathcal B(\mathbb R)$ with something like a topology?

Let me give some background to the question. In 1958, Sato famously introduced the space of hyperfunctions on $\mathbb R$ as a generalization of virtually all classical notions of functions (or distributions, or ...). His definition is very simple: It is the quotient space $$ \mathcal B(\mathbb R)=\mathcal O(\mathbb C\setminus \mathbb R)/\mathcal O(\mathbb C) $$ where $\mathcal O(U)$ denotes the space of holomorphic functions on some open subset $U\subset \mathbb C$. In other words, any hyperfunction is represented by a pair $(F^+,F^-)$ of holomorphic functions $F^+$ on the upper half-plane and $F^-$ on the lower half-plane; and two such pairs $(F^+,F^-)$ and $(G^+,G^-)$ define the same hyperfunction if $G^+-F^+$ and $G^- - F^-$ are the restriction of a single holomorphic function on all of $\mathbb C$. Naively, one may think of the hyperfunction defined by the pair $(F^+,F^-)$ as "the boundary values of $F^+$ minus the boundary values of $F^-$".

To see that this contains, at least, real-analytic functions, one notes that one can replace $\mathbb C$ above by any open subset $U\subset \mathbb C$ containing $\mathbb R$; then $$ \mathcal B(\mathbb R)\cong \mathcal O(U\setminus \mathbb R)/\mathcal O(U). $$ Now any real-analytic function on $\mathbb R$ is defined on some such $U$, and hence defines an element of $\mathcal O(U\setminus \mathbb R)$ by taking this extension of the real-analytic function on the upper half-plane, and $0$ on the lower half plane.

Any other function space carries a natural topology, but in the accounts I have looked at, the question of putting a topology on $\mathcal B(\mathbb R)$, or even just a notion of convergence, is avoided. For good reason: The inclusion $\mathcal O(\mathbb C)\hookrightarrow \mathcal O(\mathbb C\setminus \mathbb R)$ has dense image, so if we equip $\mathcal B(\mathbb R)$ with the quotient topology, it becomes indiscrete. On the other hand, I find it a bit surprising that this point is not discussed: After all, it is a function space, one wants to do functional analysis with it (like solving differential equations), so naturally one would like to talk about limits of (hyper)functions!

[Side remark, to indicate that this is a subtle point: A naive search for hyperfunctions will lead one to Graf's textbook "Introduction to hyperfunctions and their integral transforms". In this book, at the beginning of Chapter 2, Graf defines a notion of convergent sequence in $\mathcal B(\mathbb R)$, by declaring a sequence to converge in $\mathcal B(\mathbb R)$ if it can be lifted to a convergent sequence in $\mathcal O(\mathbb C\setminus \mathbb R)$. He leaves it as an exercise to the reader to show that limits are unique. Unfortunately, this is false -- as $\mathcal O(\mathbb C)\to \mathcal O(\mathbb C\setminus \mathbb R)$ has dense image by standard approximation results, any limit can be achieved.]

On the other hand, various subspaces of $\mathcal B(\mathbb R)$ have a nice topology:

  1. For any compact subset $K\subset \mathbb R$, one can define the subspace of $K$-supported hyperfunctions $\mathcal B_K(\mathbb R)\subset \mathcal B(\mathbb R)$, as the kernel of a naturally defined restriction map $\mathcal B(\mathbb R)\to \mathcal B(\mathbb R\setminus K)$ (noting that the notion of hyperfunction makes sense also for open subsets of $\mathbb R$, and yields a sheaf). Then $\mathcal B_K(\mathbb R)$ has a natural topology as a nuclear Fréchet space. In particular, the subspace of compactly supported hyperfunctions has a natural topology as a sequential colimit of nuclear Fréchet spaces.

  2. The space of distributions on $\mathbb R$ can be shown to embed naturally into $\mathcal B(\mathbb R)$. Again, the space of distributions has a nice topological structure.

The theory of condensed vector spaces handles this situation very nicely; like any quotient space, $\mathcal B(\mathbb R)$ has a natural structure as condensed $\mathbb R$-vector space and this makes the kernel of $\mathcal B(\mathbb R)\to \mathcal B(\mathbb R\setminus K)$ a condensed $\mathbb R$-vector space that comes (necessarily uniquely) from a nuclear Fréchet space.

In fact, that condensed mathematics can handle quotients by dense subspaces nicely is a thing I'm always stressing about the theory; but so far I wasn't aware of any such examples that are actually used in mathematical practice. Hyperfunctions do seem to be such a case!

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    $\begingroup$ @FedorPetrov Use the cover of $\mathbb C$ by $\mathbb C\setminus \mathbb R$ and $U$, and that the corresponding Cech complex is exact as $\mathbb C$ is Stein. $\endgroup$
    – Peter Scholze
    Commented May 10 at 6:30
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    $\begingroup$ I'm not familiar with condensed mathematics, so I don't follow your last comment. There are no compactly supported real analytic functions on $\mathbb{R}^n$ besides $0$ - more precisely, because the Taylor coefficients of any smooth, compactly supported function $f$ around any $x_0\in\partial(\text{supp} f)$ are all zero. Since the Taylor series around $x_0$ converges in a neighborhood of $x_0$ by the definition of real analyticity, we should have $f\equiv 0$. What am I missing here? Is the definition of support in condensed mathematics different from the usual one employed in real analysis? $\endgroup$
    – Pedro Lauridsen Ribeiro
    Commented May 10 at 19:16
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    $\begingroup$ @PedroLauridsenRibeiro Of course there are no compactly supported real-analytic functions. But one can still take the compactly supported cohomology of the sheaf of real-analytic functions, and it will have cohomology only in degree $1$, where one will find a nonzero $H^1_c(\mathbb R^n,\mathcal A)$ (with bad topological properties, but it's possible to consider it as condensed $\mathbb R$-vector space). Then $\mathcal B(\mathbb R^n)$ is given by $\mathrm{Ext}^1(H^1_c(\mathbb R^n,\mathcal A),\mathbb R)$. $\endgroup$
    – Peter Scholze
    Commented May 10 at 23:26
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    $\begingroup$ $\mathcal B(\mathbb R)$ is a function space, one wants to do functional analysis with it (like solving differential equations)... : There is a theorem of Hörmander about solving (constant coefficient linear) PDEs in $\mathscr D'(\Omega)/\mathscr E(\Omega)$ where solvability is characterized by $P$-convexity for singular supports. As for hyperfunctions, this space does not seem to have any reasonnable good topology since $\mathscr E(\Omega)$ is dense in $\mathscr D'(\Omega)$. Nevertheless, one can do functional analysis by a quite trivial trick... $\endgroup$
    – Jochen Wengenroth
    Commented May 13 at 12:56
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    $\begingroup$ (continued) Let $X$ be a (nice locally convex) vector space (like $\mathscr D'(\Omega)$ or $\mathcal O(\mathcal C\setminus\mathbb R)$), $L$ a subspace (possibly dense like $\mathscr E(\Omega)$ or $\mathcal O(\mathcal C)$) and $T:X\to X$ a (continuous linear) operator with $T(L)\subseteq L$. Then $T$ induces a linear map $\tilde T$ on $X/L$ which is surjective if and only if $S:X\times L\to X$, $(x,\ell)\mapsto T(x)+\ell$ is surjective. The point is that $S$ is an operator between nice spaces so that surjectivity can be investigated by functional analytic tools (for Fréchet spaces and beyond). $\endgroup$
    – Jochen Wengenroth
    Commented May 13 at 13:04

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This is a comment but will be too long. You start with the following setup: a pair of Fréchet spaces (the holomorphic functions on the complex plane, resp. on the complement of the real line there) and a natural continuous, linear injection of the former into the latter. The object of interest is then the corresponding quotient space. This is precisely the setting of the theory of quotient spaces (in your case quotient Fréchet spaces) which was introduced and studied by Lucien Waelbroeck and associates in the second half of the last century, under the name of $q$-spaces (for quotient). The setting is a suitable category from functional analysis (Banach spaces, Fréchet spaces, complete convex bornological spaces, $\dots$). One then defines a new category whose objects are pairs $E,E_0$ and a continuous injective morphism from $E_0$ into $E$. The morphisms are defined in the natural way. The objects of interest are the so-called underlying spaces $E/E_0$.

Waelbroeck began his investigations in the 60‘s but the first publication that I know of dates from 1977 („Quotient Banach spaces“, Warsaw). He considered the case of cbsˋs in the monograph „Bornological quotients“ (1998, with G. Noël). Both are easily available online.

The motivation came from operator theory—spectral theory and functional calculus.

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    $\begingroup$ Thanks!! This (e.g. Waelbroeck-Noel's book "Bornological quotients") is precisely the answer I've been looking for. It's interesting that Waelbroeck defined both "compactological spaces" which are the condensed sets that are in some sense "Hausdorff", and also worked on the problem of allowing non-Hausdorff quotients. The difference between condensed vector spaces and his considerations is that in the condensed setup, the passage from Hausdorff to non-Hausdorff objects happens before passing to vector space objects. $\endgroup$
    – Peter Scholze
    Commented May 22 at 7:01

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