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?
2026-04-08 07:13:20.1775632400
Does Hilbert–Schmidt theorem imply the space is separable?
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No. The eigenvectors are a complete orthonormal set for the image of the operator. The space would be separable if the operator were surjective.