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!



Which Banach spaces are realcompact?
Every metric space of nonmeasurable cardinality is realcompact. [1] 15.24. Thus, if there are no measurable cardinals, then every metric space is realcompact 


As you noted...To get a Banach space that is not realcompact: Let $X$ be a set with measurable cardinal, and then the discrete topology on it is not realcompact. So for example Banach space $l^1(X)$ has a closed subset homeomorphic to that, so is not realcompact.


**aside**  

There is some nice work starting with [2] on when the *weak* topology of a Banach space is realcompact (or normal, or Lindelof...). 


[1] L. Gillmann & M. Jerison, *Rings of Continuous Functions* (1960)


[2] *H. H. Corson*, [**The weak topology of a Banach space**](http://www.ams.org/mathscinet-getitem?mr=132375), *Trans. Amer. Math. Soc.* **101** (1961), 1--15.


