Orthonormal bases of Hilbert spaces

Abstract

I prove that a Hilbert space has the property that each of its dense (not necessarily closed) subspaces contains an orthoormal basis if and only if it is separable.

The main result of the note has been known. See Orthonormal bases and quasi-splitting subspaces in pre-Hilbert spaces D. Buhagiar, E. Chetcuti, H. Weber, J. Math. Anal. Appl., 345 (2008) 725-730. and the references thereof. Thanks to Nigel Kalton to pointing this out