What is an example of a non-paracompact topological vector space?


I'm aware of [this question](https://mathoverflow.net/questions/3241/when-is-a-locally-convex-topological-vector-space-normal-or-paracompact), but I don't care if my tvs is locally convex. In fact the wilder the better. The only criterion is that it should be contractible, which it would be, assuming $\mathbb{R} \times V \to V$ is continuous.


This is in order to answer [this other question](https://mathoverflow.net/questions/44859/cw-complexes-and-paracompactness).



An example of a non-paracompact tvs (over the reals, say)
Henno Brandsma provided this answer in a comment to anon's answer, which I think is the easiest and best, hence I'm adding it as a community wiki answer.


Consider an uncountable set $\mathfrak{n}$ and the space $\mathbb{R}^\mathfrak{n}$ in the product topology. This is [not normal](http://www.ams.org/journals/bull/1948-54-10/S0002-9904-1948-09118-2/home.html), but still Hausdorff, hence not paracompact.


