Basis of infinite dimensional Banach space and separable hilbert space

Question

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.

Answer 1

An "orthonormal basis" is not the same thing as a basis that is orthonormal, and in particular may not even actually be a basis at all. An orthonormal basis for a Hilbert space $H$ is just a maximal orthonormal subset $B\subseteq H$. Such a subset need not actually span $H$, and so it need not be a basis. It is true that we can write an arbitrary element $x\in H$ as the sum $\sum_{b\in B}\langle x,b\rangle b$, but this is an infinite sum in general and so does not show that $x$ is in the span of $B$.

So, a separable infinite dimensional Hilbert space has a countable orthonormal basis, but no such orthonormal basis is actually a basis so this does not contradict your Theorem 1.