Setup: we consider the category of locally convex topological vector spaces with morphisms as continuous linear maps. This time, I'm explicitly allowing the axiom of choice (or at least the Hahn-Banach Theorem) - for reasons that will become obvious in a moment. The space of continuous linear maps from one LCTVS to another can be given a variety of structures as an LCTVS, for example:

- Uniform convergence on bounded sets (strong topology)
- Uniform convergence on relatively compact sets
- Uniform convergence on compact sets
- Uniform convergence on finite sets (simple topology)
- Pointwise convergence on each of the above (varieties of weak topologies)

These examples are all functorial in both source and target.

Question: Are there any other functorial topologies?

Background: One day, gathered round a table in the back room of the n-category cafe, Todd Trimble and I (with occasional help from others) were talking about monoidal structures on the category of LCTVS (we were meant to be talking about smooth spaces). We got quite far and found several such structures which differ essentially just up to topology. In particular, there's a monoidal structure (even symmetric) for each of the **uniform** topologies above. And if anyone comes up with another functorial topology based on uniform convergence, there'll be a monoidal structure for that. The discussion stalled a little towards the end and I'd quite like to finish and see if it's possible to classify all monoidal structures on LCTVS.

Incidentally, the HBT is important in this because it allows one to show that the unit of a monoidal structure on LCTVS has to be $\mathbb{R}$. The unit has to satisfy the property that $\dim \mathcal{L}(X) = 1$. With HBT, this is just $\mathbb{R}$. Without HBT, one could take $\mathbb{R} \oplus \ell^\infty/c_0$.