$H$ is a Hilbert space , $M_0$ and $M_1$ are two orthonormal basis of $H$ , can we prove that $M_0$ and $M_1$ have the same cardinality ?
My attempt :
If $M_0$ is infinite and countable $M_1$ is infinite , then we can find a countable dense subset of $H$ , then for each element $g \in M_1$ , consider the ball centered at $g$ with radius $\frac12$ , we can conclude that $M_1$ is countable . So $M_0$ and $M_1$ have the same cardinality.
If we only assume $M_0$ and $M_1$ are infinite , I want to prove the conclusion by the similar way if we can prove the following statement :
Let $E$ denote an infinite linear space , then the set $$E_1=\{f | f=a_1e_{\alpha}+...+a_n e_{\gamma} , a_i \in Q \, , e_{\alpha} \in E \text{ and } n \in Z^+ \}$$
has the seem cardinality as $E$ .
If $E_1$ is countable , then I know that countable union of countable set is still countable . But is it still true for $E_1$ is uncountable ?