Question

Every Hilbert space has an orthonormal basis.

Answer 1

The theorem states, that every Hilbert space contains a total orthonormal set. Furthermore, all total orthonormal sets in a Hilbert space H =/= {0} have the same cardinality, which is known as the Hilbert dimension. The proof requires the axiom of choice or Zorn’s lemma.

So, the statement is correct.