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).