A paper dealing with this (and other) questions for the *weak topology of a Banach space* ... H. H. Corson, "The weak toplology of a Banach space" Trans. Amer. Math. Soc. 101 (1961) 1--15.

In that case:

X is paracompact iff X is Lindelof.

If X^n is normal for all n, then X is real-compact.