fa.functional analysis - When is a locally convex topological vector space normal or paracompact?

Thanks to your other question, I was on a LCTVS kick. I did find one general criterion that implies that a locally convex space is paracompact. According to the Encyclopedia of Mathematics, if it is Montel (which means that it is barrelled and the Heine-Borel theorem holds true for it), then it is paracompact. Although this criterion is important, it is of no use to your specific question.

I thought I had a proof for half of your question, which I wrote up as the first version of this answer, but I made a mistake and proved something different. My thinking is based on the fact that the normality axiom for a topological space is equivalent to the Tietze extension theorem. (Tietze extension follows from normality. In the other direction, if $A$ and $B$ are the two closed sets, you obtain disjoint open neighborhoods from a continuous function that is 0 on $A$ and 1 on $B$.) However, in my argument I conflated the locally convex direct sum of spaces with the topological direct sum. For a countable direct sum of copies of $mathbb{R}$, they are the same topology, and they agree with the box topology. But Waelbroeck, LNM 230 points out that they are different in the uncountable case.

Let $alpha$ be an ordinal, for instance an ordinal of cardinality $2^{aleph_0}$. Then $mathbb{R}^alpha$ in the topological direct sum topology satisfies Tietze extension. Let $A subset mathbb{R}^alpha$ be a closed set and let $f:A to mathbb{R}$ be a continuous function. For $beta < alpha$, let $A_beta$ be the intersection of $A$ and with $mathbb{R}^beta$. Suppose that $alpha = beta+1$ is a successor ordinal. If $alpha$ is finite, then the conclusion is standard. Otherwise, by induction, there is an extension $f_beta$ of $f$ to $mathbb{R}^beta$. Moreover, by induction in a different sense, we have already proved that $mathbb{R}^{beta+1}$ is normal, since $beta$ and $beta+1$ have the same cardinality. So there exists an extension $f_alpha$ to $mathbb{R}^alpha$. If instead $alpha$ is a limit ordinal, then the extensions all the way up to $alpha$ work just because they work; that's the behavior of topological direct limits.

Having failed to normality for the locally convex direct sum, I can't say much about paracompactness either. :-) However, there is an interesting result called the Michael selection theorem which seems to do for paracompactness what the Tietze theorem does for normality. If the Tietze theorem is useful for your spaces, then maybe the Michael selection theorem is too.