Topology on $O_M$, the space of slowly increasing smooth functions?

Question

A smooth function on $\mathbb{R}^n$ is called slowly increasing if each of its derivatives is polynomially bounded. It seems that the collection of such functions is denoted as $O_M$.

Obviously, $O_M$ is a vector subspace of $\mathcal{S}'(\mathbb{R}^n)$ and a multiplicative subalgebra of $C^\infty(\mathbb{R}^n)$.

I am aware that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is a nuclear (tamed) Fréchet space, so that $\mathcal{S}'(\mathbb{R}^n)$ is also nuclear with the strong dual topology. Also, topology of uniform convergence of all derivatives on any compact set is a Fréchet topology on $C^\infty(\mathbb{R}^n)$ but I don't know much of the details. Moreover, I have difficulty finding a detailed description of a topology on $O_M$.

So, I would like to ask here two questions.

Is $C^\infty(\mathbb{R}^n)$ also nuclear as a Fréchet space? The Wiki page on Montel spaces says that $C^\infty(\mathbb{R}^n)$ is Montel, and I am aware that a nuclear Fréchet space is Montel. So, I guess that I am correct?

Exactly what topology is "natural" on $O_M$? Perhaps it can be give some LF topology which is hopefully nuclear and also Montel?

Could anyone please provide any information or reference? I will split this post into two if that makes more sense.

Answer 1

I think $C^{\infty}(\mathbb{R}^n)=\mathscr{E}(\mathbb{R}^n)$ is indeed nuclear, if I remember well. The reference to check for that would be the book by Trèves "Topological Vector Spaces, Distributions and Kernels".

For $\mathscr{O}_M(\mathbb{R}^n)$, the standard topology is the locally convex one defined by the (uncountable) collection of seminorms $$ f\mapsto\rho(gf) $$ where $g$ ranges over Schwartz space $\mathscr{S}(\mathbb{R}^n)$, and where $\rho$ ranges over continuous seminorms on $\mathscr{S}(\mathbb{R}^n)$. The topology of $\mathscr{O}_M(\mathbb{R}^n)$ is rather subtle. This space was conjectured to be isomorphic to $s\widehat{\otimes}s'$ by Grothendieck. Here $s$ is the space of rapidly decreasing sequences and $s'$ is the dual space of sequences of at most polynomial growth. This conjecture was proved about thirty years later by Valdivia. A good reference for $\mathscr{O}_M(\mathbb{R}^n)$ is the book by Horváth "Topological Vector Spaces and Distributions".

Also, a good reference for the topologies of all these spaces is the relatively recent article by Christian Bargetz "Explicit representations of spaces of smooth functions and distributions" Journal of Mathematical Analysis and Applications, vol. 424, no. 2, 1491--1505, 2015.