Hellinger–Toeplitz theorem

The Hellinger–Toeplitz theorem states that an everywhere-defined symmetric linear operator on a Hilbert space is bounded and therefore continuous.

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Symmetric operator: A linear operator on an inner-product space whose inner products satisfy ⟨Tx,y⟩ = ⟨x,Ty⟩ on its domain. The theorem’s hypothesis is precisely symmetry, together with an operator defined on the whole Hilbert space.
Inner product: A function that assigns scalars to pairs of vectors and induces notions of length and orthogonality. Symmetry is expressed through inner products of operator outputs and test vectors.
Unbounded operator: An operator that has no finite constant bounding its output norm by its input norm. The theorem excludes unboundedness when symmetry and an everywhere-defined domain both hold.
Quantum observable: A measurable physical quantity represented in quantum theory by a self-adjoint operator. The theorem explains why unbounded observables cannot be represented by everywhere-defined symmetric operators.
Closed graph theorem: A linear map between Banach spaces is bounded if its graph is closed. Symmetry lets convergence of inputs and outputs force the graph of the operator to be closed.
Dense domain: A subspace of a normed space whose closure is the entire space. An everywhere-defined operator has a dense domain, but density alone does not yield the theorem’s conclusion.
Multiplication operator: An operator that multiplies functions pointwise by a fixed function, subject to a specified domain. Multiplication by an unbounded function gives familiar unbounded operators when the domain is restricted.
Spectral theorem: A family of results representing suitable operators through spectral data such as eigenvalues or projection measures. Boundedness helps identify when elementary bounded-operator spectral theory applies.
Graph of a linear operator: The set of pairs consisting of an operator’s input and its output, viewed as a subset of the product space. The proof establishes that this set is closed before applying the closed graph theorem.
Adjoint operator: The operator defined through the inner-product identity relating an operator to its action on test vectors. Comparing symmetry with the adjoint clarifies why domain assumptions matter.