Does the definition of Schauder basis of a Banach space include that it be countable.
My issue is a theorem that states a Hilbert space is separable if and only if it has a countable orthonormal basis.
Now if countable is part of the definition of Schauder basis, then why include it in the statement of the theorem? If countability is part of the definition of a Schauder basis then shouldn’t the statement just be ‘a Hilbert space is separable if and only if it has a Schauder basis’ since for any Schauder basis can be made into an orthonormal one by the gram Schmidt process.