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.

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.