# Does Hilbert–Schmidt theorem imply the space is separable?

The Hilbert–Schmidt theorem says a self-adjoint compact operator on a Hilbert space have a complete orthonormal set consisting of eigenvectors. Does that imply the space is separable?

• $0$ is a compact operator on any Hilbert space. A Hilbert space $H$ is separable iff there exists exists an injective compact selfadjoint operator $A$ on $H$. – DisintegratingByParts Nov 7 '14 at 18:40