George Jones said:The [itex]T[/itex] that I defined is an inner product space that needs to be completed (made "closed" with respect to Cauchy sequences) in order to obtain a Hilbert space.
Up to here I follow. However, you probably agree that this closure adds (a LOT) of elements to the set, ....
Ah, that's perhaps the key: you have at least a DENSE, countable subset: namely your basis vectors, say, with rational coefficients.
I vaguely remember that there is a theorem that says that a hilbert space that has a countable dense subset is separable, is that the point ?
EDIT: I'm beginning to see the point. The basis of the closure is NOT the closure of the basis (which is uncountable)