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 (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...

, a branch of mathematics, the Hellinger–Toeplitz theorem states that an everywhere-defined symmetric operator on a Hilbert space with inner product

\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, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a st ...

, 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 In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. Graphs and m ...

, as self-adjoint operators are closed. Alternatively, it can be argued using the uniform boundedness principle. 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, which ...

Observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...

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. 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 of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...

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