# Is every locally Banach, Hausdorff space regular?

I am working on some infinite dimensional differential geometry. I have tried proving a somewhat weaker statement than the above by replacing locally Banach with locally metrizable. But after some research this is obviously wrong as the K-topology is a contradiction to such a statement, see A locally metrizable, Lindelöf Hausdorff space that is not metrizable.

My goal is to use this to prove that second countable locally Banach, Hausdorff spaces are metrizable, which is equivalent to locally metrizable, Hausdorff and paracompactness. That is to prove that we need not worry about convention used to define Banach manifold. Does anyone have a reference for an article or such that treats these questions? I have looked into some books on Banach manifolds but without any luck.

If it is not the case that every locally Banach, Hausdorff space is regular, then does anyone have a counterexample at hand? (If this is the case the article I am trying to use would be incorrect on this point).

But if we weaken paracompactness or omit it altogether, while strengthening locally metrisable to locally Banach, I'm not so sure what happens: in the finite-dimensional (locally Euclidean) case, things are easy: we are locally compact and so Hausdorffness implies Tychonoffness (stronger than regular). But the infinite dimensional case will be nowhere locally compact. I don't see a handy theorem to guarantee regularity. (If $$X$$ is even second countable, the regularity gives paracompactness and then Smirnov applies)