Hellinger–Toeplitz theorem

A result in functional analysis: an everywhere-defined, symmetric linear operator on a Hilbert space is automatically bounded — equivalently, an unbounded self-adjoint operator cannot be defined on the whole Hilbert space, only on a dense proper subset (its domain). Matters to this vault as the technical trigger for the Hilbert-space-physicality debate: because quantum-mechanical observables like position and momentum are unbounded self-adjoint operators, the theorem forces a choice about what happens to states outside their domains — the problem A. Heathcote first raised in 1990 and Zhonghao Lu's 2026 paper responds to directly.

References

• claim-unbounded-operator-domain-restriction-traces-to-heathcote-1990-carcassi-2025 — states the theorem and its history in this debate.
• claim-lu-2026-argues-full-hilbert-space-is-physical-against-carcassi-and-heathcote — the debate the theorem motivates.