In functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Hellinger–Toeplitz theorem states that an everywhere defined symmetric operator on a Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

 is bounded

Bounded operator

In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...

. By definition, an operator A is symmetric if
for all x, y in the domain of A. Note that symmetric everywhere defined operators are necessarily self-adjoint

Self-adjoint operator

In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...

, so this theorem can also be stated that an everywhere defined self-adjoint operator is bounded. The theorem is named after Ernst David Hellinger and Otto Toeplitz

Otto Toeplitz

Otto Toeplitz was a German Jewish mathematician working in functional analysis.- Life and work :...

.

This theorem can be viewed as an immediate corollary of the closed graph theorem

Closed graph theorem

In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph.- The closed graph theorem :...

, as self-adjoint operators are closed

Closed operator

In mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the...

. Alternatively, it can be argued using the uniform boundedness principle

Uniform boundedness principle

In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field...

. One relies on the symmetric assumption, therefore the inner product structure, in proving the theorem. Also crucial is the fact that the given operator A is defined everywhere (and, in turn, the completeness of Hilbert spaces).

The Hellinger–Toeplitz theorem leads to some technical difficulties in the mathematical formulation of quantum mechanics

Mathematical formulation of quantum mechanics

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...

. Observable

Observable

In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...

s in quantum mechanics correspond to self-adjoint operators on some Hilbert space, but some observables (like energy) are unbounded. By Hellinger–Toeplitz, such operators cannot be everywhere defined (but they may be defined on a dense subset). Take for instance the quantum harmonic oscillator

Quantum harmonic oscillator

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

. Here the Hilbert space is L

Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

²(R), the space of square integrable functions on R, and the energy operator H is defined by (assuming the units are chosen such that ℏ = m = ω = 1)

This operator is self-adjoint and unbounded (its eigenvalues are 1/2, 3/2, 5/2, ...), so it cannot be defined on the whole of L²(R).