Let $O\_M(\mathbb{R}^n):= \mathcal{S}'(\mathbb{R}^n) \cap C^\infty(\mathbb{R}^n)$ be the space of slowly increasing smooth functions on $\mathbb{R}^n$.


Following p.294 proposition 9.10 of the "Real Analysis" (1999) by Gerald Folland, define the convlution map $\mathcal{S}'(\mathbb{R}^n) \times \mathcal{S}(\mathbb{R}^n) \to O\_M(\mathbb{R}^n)$ by
\begin{equation}
(F, \psi) \to F \* \psi \text{ where } (F \* \psi)(x):= \langle F, \tau\_x\psi \rangle
\end{equation}


Then, the answer in the link
[Can distribution theory be developed Riemann-free?](https://mathoverflow.net/questions/72450/can-distribution-theory-be-developed-riemann-free/351028#351028)


says that this bilinear map is in fact "hypocontinuous". I found the definition of hypocontinuity myself on Wikipedia.


However, what topology is imposed on $O\_M(\mathbb{R}^n)$ in this case? Is it the subspace topology inherited from $\mathcal{S}'(\mathbb{R}^n)$? Or is it something else?


I am quite confused...Could anyone please clarify?



What exactly is the topology on $O_M$ that makes the convolution map $S \times S' \to O_M$ hypocontinuous?
The definition is wrong. For example in 1d, the function $\sin(e^x)$ is in the intersection of $S'$ and $C^{\infty}$ but is not in $O\_M$ which is a strict subset (all 3 spaces being seen as subsets of $D'$ otherwise the intersection does not even make sense).


The definition (for the real-valued case) is the space of $C^{\infty}$ functions $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that
$$
\forall \alpha\in \mathbb{N}^n, \exists k\in\mathbb{N}, C\ge 0,\forall x\in\mathbb{R}^n,
|\partial^{\alpha}f(x)|\le C\langle x\rangle^k\ .
$$
I used the standard multiindex notation for derivatives, and the notation for the Japanese bracket $\langle x\rangle=\sqrt{1+|x|^2}$ where $|x|$ is the Euclidean norm. Also my $\mathbb{N}$ starts with 0.


The only topology safe to use on $O\_M$, i.e., the canonical one given its characterization as a multiplier space, is the locally convex topology defined by the seminorms
$$
f\longmapsto\|g f\|
$$
where $g$ ranges over all Schwartz functions in $S$ and $\|\cdot\|$ ranges over all
continuous seminorms of $S$.


Note that in my answer referred to in the question, I did give the definition of the topology. I also succintly explained how to go about learning the proofs of distributional Fubini a)-d): use the isomorphism with sequence spaces provided by Hermite functions which tensorize well, then do it by hand in the sequence space situation. For definitions and crieteria of continuity and hypocontinuity, see Wikipedia for hypocontinuity and


<https://math.stackexchange.com/questions/3510982/doubt-in-understanding-space-mathscr-d-omega/3511753#3511753>


for continuity.


To learn more, most distribution theory textbooks written for a PDE audience might not be very helpful, i.e., they may not even mention $O\_M$. You will need a book that sticks more closely to the theory built by Laurent Schwartz.
There is the book by Schwartz himself (only in French), the one by Horváth or the one by Trèves. If you can find them there are also nice lectures by Amann.


