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.