A paper dealing with this (and other) questions for the weak topology of a Banach space ... H. H. Corson, "The weak topology of a Banach space" Trans. Amer. Math. Soc. 101 (1961) 1--15.
In that case:
$X$ is paracompact iff $X$ is Lindelöf.
If $X^n$ is normal for all $n$, then $X$ is real-compact.
