Let \(H\) denote a Hilbert space . The following statements are equivalent [1]:

1. \( H \) is separable .
2. All orthonormal bases of \( H \) are countable.
3. There exists at least one countable orthonormal basis.

References Link to heading

1. D. Werner, Funktionalanalysis. Berlin, Heidelberg: Springer Berlin Heidelberg, 2018. doi:10.1007/978-3-662-55407-4