I have a question about the topological space underlying a Banach space.

A topological space $X$ is realcompact iff it is homeomorphic to a closed subset of an infinite product of the form $\mathbb R^\kappa$. Closed subsets of realcompact spaces are realcompact.

A classical result in infinite topology states that every infinite dimensional *separable* Banach space is hemeomorphic to $\mathbb R ^ \omega$, so in particular, a separable Banach space is realcompact.

What about non-separable Banach spaces? Are they realcompact?

Since it is consistent with ZFC that there are discrete space which are not realcompact and every discrete set $X$ is a closed discrete subset of a space $\ell^p(X)$, it seems that in general the answer will be no. But of course, this counterexample is a huge space and in particular, it is not a ZFC-counter-example.

So, one question would be: Is there a ZFC-example of a Banach space which is not realcompact? Another question would be: Is there an easy way to determine which Banach spaces are realcompact and which are not?

Thank you!