A Hilbert space is called separable if it has a dense, countable subset, which occurs iff it admits a countable orthogonal basis. Separability in the topological sense is the same as saying the topological space $(X,\mathcal{T})$ has a countable base.
Since all Hilbert spaces are equipped with an inner product, which can be used to define a metric via $d(f,g) = (f-g,f-g)^{1/2}$, then one call always define a metric topology on any Hilbert space. Let us call this the nominal metric topology on a Hilbert space $X$.
Now suppose $X$ is a separable Hilbert space. Is the topological space $(X,\mathcal{T})$ necessarily separable in the topological sense when $\mathcal{T}$ is the nominal metric topology on $X$? What about if $\mathcal{T}$ is another topology? (some easy examples clearly fail--the discrete topology, for instance)
