Is there a collectionwise normal topological vector space which is not paracompact?

Do you know a collectionwise normal topological vector space that is not paracompact?

• I am looking for an example of a collectionwise normal topological vector space that is not paracompact. Any idea about it?

• Answer:

  Note: This is wrong, because I managed to miss vector space when reading the question. I’m leaving it for now to make sure that the OP sees my last comment. One of the simplest examples is $\omega_1$, the first uncountable ordinal (viewed as the set of all countable ordinals) with the order topology; for a discussion of some of its properties see http://dantopology.wordpress.com/2009/10/11/the-first-uncountable-ordinal/ to Dan Ma’s Topology Blog. It’s a linearly ordered space with the order topology, so it’s hereditarily collectionwise normal, but it’s not paracompact. For another proof that $\omega_1$ is collectionwise normal see http://dantopology.wordpress.com/tag/collectionwise-normal/ to the same blog. http://math.stackexchange.com/users/4280/henno-brandsma answer to http://mathoverflow.net/questions/78975/ccc-collectionwise-normality-paracompact gives another example.