Functions of moderate increase compactly generated? Let $\mathcal{O}_M(\mathbb{R}^d)$ be the space of smooth moderately increasing functions
$\{ f \in \mathcal{C}^\infty(\mathbb{R}^d) : \forall \alpha \exists N \text{ such that} \Vert \langle \cdot \rangle^{-N} \partial^\alpha f \Vert_{L^\infty} < \infty \}.$
This space is equipped with the locally convex topology generated by the seminorms $\rho_{\phi, \alpha}(f) : = \Vert \phi \partial^\alpha f \Vert_{L^\infty}$ for $\phi \in \mathcal{S}(\mathbb{R}^d)$ and $\alpha$ a multi-index. Is the space $\mathcal{O}_M(\mathbb{R}^d)$ a compactly generated space?
Various properties of this space and its relation to $\mathcal{S}(\mathbb{R}^d)$ were proved by Schwartz, and Grothendieck proved in his thesis that the space is bornological.
So far, I cannot show this space is compactly generated. Here is what I have found so far:

*

*Locally compact spaces are compactly generated, which is not useful here since any infinite dimensional Hausdorff topological vector space is not locally compact.

*If the space was metrizable, then it would be compactly generated. However, I do not know if the uncountable set of seminorms above can be reduced to a countable one giving the same topology.

*Bornological spaces can be written as a direct limit of Banach spaces, but a direct limit of compactly generated spaces is not necessarily compactly generated.

 A: Not an answer but a simplification of the question.
Let $s$ be the space of sequences $x=(x_n)_{n\ge 0}$ in $\mathbb{R}^{\mathbb{N}}$ such that for all $k\in\mathbb{N}$
$$
||x||_k:=\sup_{n\ge 0}\ (n+1)^{\frac{k}{2}}|x_n|<\infty\ .
$$
Give $s$ the locally convex topology defined by the seminorms $||\cdot||_k$, $k\ge 0$. Namely, $s$ is the just the space of sequences of rapid decay.
Let $s_{+}$ be the subset of $s$ made of sequences with nonnegative entries.
Let $s'$ be the space of sequences of temperate (at most polynomial) growth.
Let $s'_{+}$ be the subset of $s'$ made of sequences of with nonnegative entries.
Let $\mathcal{M}$ be the spaces of "matrices" $a=(a_{i,j})_{(i,j)\in\mathbb{N}^2}$
such that
$$
||a||_{\omega,\omega'}:=\ 
\sum_{(i,j)\in\mathbb{N}^2}\omega_i \omega'_j |a_{i,j}|
<\infty
$$
for all $\omega\in s_{+}$ and all $\omega'\in s'_{+}$.
Give $\mathcal{M}$ the locally convex topology defined by the seminorms $||\cdot||_{\omega,\omega'}$.
If I remember correctly, Grothendieck conjectured that $\mathscr{O}_{\rm M}$ is isomorphic to $\mathcal{M}$ and this was proved some thirty years later by Valdivia.
Since the question is about the topological vector space structure of $\mathscr{O}_{\rm M}$ only, it might be easier to examine it for $\mathcal{M}$ instead.
