It is known that an infinite dimensional Banach space does not have a countable Hamel basis. It is also known that a separable Hilbert space has an orthonormal countable basis. Now, I think this basis is Schauder basis. So this means that a Banach space can have uncountable Hamel basis but a countable Schauder basis, right? Or am I missing something here?
By the way, do separable Banach(non-Hilbert) spaces also have countable Schauder bases? Thanks beforehand.