Orthonormal bases of Hilbert spaces

TL;DR

This paper proves that a Hilbert space's dense subspaces contain orthonormal bases if and only if the space is separable, highlighting a key property related to the structure of Hilbert spaces.

Contribution

It establishes a necessary and sufficient condition for dense subspaces of Hilbert spaces to contain orthonormal bases, linking this property to separability.

Findings

Dense subspaces contain orthonormal bases iff the space is separable

Separable Hilbert spaces have this property for all dense subspaces

Non-separable spaces lack this property in some dense subspaces

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.