Functions of moderate increase compactly generated?

Question

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:

1. Locally compact spaces are compactly generated, which is not useful here since any infinite dimensional Hausdorff topological vector space is not locally compact.
2. 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.
3. 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.

Answer 1

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.