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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

with

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...

\langle \cdot ,  \cdot \rangle

is bounded. By definition, an operator ''A'' is ''symmetric'' if :

\langle A x ,  y \rangle = \langle x ,  A y\rangle

for all ''x'', ''y'' in the domain of ''A''. Note that symmetric ''everywhere-defined'' operators are necessarily

self-adjoint In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a^*). Definition Let \mathcal be a *-algebra. An element a \in \mathcal is called self-adjoint if The set of self-adjoint elements ...

, so this theorem can also be stated as follows: an everywhere-defined self-adjoint operator is bounded. The theorem is named after Ernst David Hellinger and Otto Toeplitz. This theorem can be viewed as an immediate corollary of the closed graph theorem, as self-adjoint operators are closed. Alternatively, it can be argued using the

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 cornersto ...

. 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 reveals certain technical difficulties in the

mathematical formulation of quantum mechanics The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, whic ...

Observable In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum ...

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 In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...

). Take for instance the

quantum harmonic oscillator The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...

. Here the Hilbert space is L²(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) :

x) = - \frac12 \frac f(x) + \frac12 x^2 f(x).

This operator is self-adjoint and unbounded (its

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

s are 1/2, 3/2, 5/2, ...), so it cannot be defined on the whole of L²(R).

References

* Reed, Michael and Simon, Barry: ''Methods of Mathematical Physics, Volume 1: Functional Analysis.'' Academic Press, 1980. See Section III.5. * {{DEFAULTSORT:Hellinger-Toeplitz Theorem Theorems in functional analysis Hilbert spaces