I got two theorems.
Theorem 1. Infinite dimensional Banach space has uncountable basis( cannot have countable basis).
Theorem 2. Infinite dimensional Hilbert space is separable iff every orthonormal basis for the Hilbert space is countable.
And notice that Hilbert space is Banach space and it means that every infinite dimensional Hilbert space has uncountable basis.
In my mind, they are contradicting each other. I don't know where I am thinking wrongly. I hope to get any help from here. Any comment would be helpful for me.
Thank you in advance.