Context: I'm trying to find out which topological (unital) modules are "good enough" for generalizing results from real or complex functional analysis. For example, I say that a module, in order to be "good enough", must have a basis, and also its submodules. But that fact is equivalent to use a division ring, so this first problem is solved.
My problem is when I want to find out if a module has or not "enough continuous linear functionals". The most frequent version of Hahn Banach implies that there's a lot of them for real or complex normed spaces. A more general version (e.g, on Topological Vector Spaces I, Köthe, p.233) finds enough functionals for real or complex locally convex spaces. Here arises my first question:
(1) On non locally convex spaces, are there nontrivial functionals? I know that $L^p$ for $0<p<1$ has none but $0$, but that is just an example.
Another problem arises. I need the notion of convexity to define locally convex spaces, but in modules that is not possible in general. Demanding the ring to be absolutely valued would solve the problem, but that is too strong (in fact, I think that condition restricts me to the first four Cayley Dickson algebras...). So I'm thinking about using this notion
http://en.wikipedia.org/wiki/Convex_metric_space
to make a "locally convex metric module" version of Hahn Banach, which is weaker (although maybe I'll need to demand completeness). If this works, will solve part of my problem, but there are still a lot of modules out there. So, my other questions are:
(2) What can I do where I can't define convexity, or something similar that would allow me to use Hahn Banach?
(3) Can you recommend me some bibliography on convex metric spaces? I found almost nothing on google and I need some written help.
Thanks in advance,
Pablo