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.