---
title: "Hellinger–Toeplitz theorem"
type: "entity"
entity_kind: "concept"
status: "hub"
canonical_name: "Hellinger–Toeplitz theorem"
aliases: ["Hellinger-Toeplitz theorem"]
first_seen: "2026-10-05T00:00:00.000Z"
writer_model: "claude-sonnet-5"
connects_to: ["unbounded operators","Hilbert space physicality","self-adjoint operators","Zhonghao Lu","A. Heathcote"]
seek_code_commit: "379c8e7"
---


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 [[entity-a-heathcote-philosophy-of-science|A. Heathcote]] first raised in 1990 and [[entity-zhonghao-lu|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.