# Uncountable basis in Hilbert space vs orthonormal basis

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.

• First question: yes. The "by the way": If a Banach space has a Schauder basis, then it is countable. (Not all separable Banach spaces have a Schauder basis.) – David Mitra May 4 at 15:30
• @DavidMitra yes, I just saw that P. Enflo had given an example of a separable Banach space wiout a Schauder basis – vidyarthi May 4 at 15:35