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](https://doi.org/10.1090/S0002-9947-1961-0132375-5 )" 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.