All locally convex topological vector spaces (LCTVS) are completely regular, since their topology is given by a family of semi-norms. I'm interested in conditions that imply that a LCTVS is paracompact or normal.

Some background: I have a particular space in mind. It is a directed colimit (union) over an uncountable family of nuclear Frechet spaces. It is complete but not metrisable nor separable nor nuclear. I have a very concrete description of it and can describe the bounded (and compact) subsets. This space is fairly badly behaved in other ways so I'm half anticipating a negative result, thus answers along the lines of "If you can find a subset that looks like X then it can't be normal" could be just what I'm looking for.

So in particular, I'm interested in the general question. To forestall a couple of "easy" answers: as my space is not Frechet, it is not metrisable so I can't directly use theorems on metrisability (however, as it is a colimit there may be some scope for indirect use). And, of course, paracompact would imply normality since it is completely regular.

To forestall another possible comment, I'm not going to tell you what the particular space is. Partly because I'm more interested in the general situation, this space is merely focussing my attention on the question, and partly because it's a more useful question if it's general. A bit of intelligent searching would reveal what space it is anyway so it's no great hardship.

**Edit:** There is a simple example of a space that would be very interesting to know about: the sum (i.e. coproduct) of an uncountable number of copies of $\mathbb{R}$, or even more specifically $\sum_{\mathbb{R}} \mathbb{R}$. This isn't the specific space that I'm interested in, but is close enough that I think that an answer either way for this space will tell me what to do for my space.

locally convextopology so that the inclusions of allfinitesubsums are continuous. $\endgroup$ – Andrew Stacey Nov 13 '09 at 10:28