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.

  • 1
    $\begingroup$ 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.) $\endgroup$ – David Mitra May 4 at 15:30
  • $\begingroup$ @DavidMitra yes, I just saw that P. Enflo had given an example of a separable Banach space wiout a Schauder basis $\endgroup$ – vidyarthi May 4 at 15:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.