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Suppose we have a separable Hilbert space $H$ having a countable orthonormal basis $\{e_i\}$. How do I show that any orthonormal set of vectors has to be at most countable?

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If $\{e_i\}_{i \in I}$ is a ONS, you have $$\| e_i - e_j \| = \sqrt{2}$$ for $i \ne j$. Since $H$ is separable, you have that $I$ is at most countable.

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