bornological vector spaces over a non-archimedean field

Question

Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in 'Espaces analytiques relatives et theorem de finitude' by Houzel it is assumed that the valuation is non-discrete and that $k$ is maximally complete. On page 43 of 'Seminaire Banach' published as Springer Lecture notes in Mathematics volume 277, it is assumed that the valuation is non-discrete. What is the main reason behind these restrictions? I am interested in bornological vector spaces over a field with trivial valuation. Banach spaces over such a field make sense and therefore one gets a metric space and therefore a bornological set where the bornology is compatible with the linear structure. This should still be a (complete) bornological vector space hopefully. For what parts of the theory are these extra restrictions needed or useful? Are there some pathologies about the category of bornological vector spaces over a general complete non-archimedean field that are not present when you add these extra assumptions?

Answer 1

I try to reply to your questions:

1) In the first part of "Espaces analytiques relatives et theorem de finitude" Houzel says that the field under consideration is supposed to be maximally compact. I don't see any point where he use this hypothesis in his article neither in the result that he recall from the "Seminarie Banach". Moreover, in the second part, where he expose the sheaf-theoretic (global) version of the result obtained in the first part he recall the notation and remove this hypothesis (see on page 29 the first two lines of the paragraph "Faisceaux bornologiques"). Hence my idea with respect to this hypothesis is that it is simply a misprint.

2) For sure you can develop a theory of bornological or topological vector spaces over any valued field (also over any field with more exotic structures). The problem is that already in the case of trivially valued fields you find pathologies. First, the notion of convexity is quite strange: since $k = k^\circ$ then you find that the natural generalization of the absolute convex hull of a subset $X \subset E$ (where $E$ is a $k$-vector space) is given by $\Gamma(X) = \left \{ \sum \lambda_i x_i | \lambda_i \in k^\circ = k, x_i \in X \right \}$. Hence $\Gamma(X)$ is the linear span of $X$.

The main problem in this situation is that there is no analog of the duality between bornological and topological vector spaces, which is the main topic of the seminaire Banach. Houzel construct two adjoint functors $t: Born \to Top$ and $b: Top \to Born$ where $Born$ is the category of bornological vector spaces of convex type and $Top$ is the category of locally convex topological vector spaces. For the $t$ functor you consider on a bornological vector space of convex type the vector space topology given by bornivorous sets (i.e. sets which absorbs all bounded sets). And for the $b$ functor you consider on locally convex topological vector space the Von Neumann bornology.

So if you perform this construction on a seminormed vector space $E$ over a non-trivially valued field you get that $E \cong b(t(E))$ and $E \cong t(b(E))$ (but this is also true for more general $k$-vector space topologies and bornologies). This is false for the trivial valued field even thinking of $k$ as a $1$-dimensional vector space over itself. Cause if $k$ is trivially valued, then all subsets of $k$ are bounded. In particular $k$ itself is bounded, so the only bornivorus set for this bornology is $k$. Therefore, in this case, $t(k)$ has the indiscrete topology but the trivial valuation gives to $k$ the discrete topology. Moreover, if you try to describe the Von Neumann bornology for the discrete topology, you don't find a bornology cause $0$ is a neighborhood of itself and $0$ doesn't absorb nothing else than itself. This mean the $0$ would be the only bounded set for the Von Neaumann bornology but this doesn't make sense.

This lack of duality is (i think) the main issue for which Houzel exclude the trivial valuation in his work.