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- Hellinger–Toeplitz_theorem abstract "In functional analysis, a branch of mathematics, the Hellinger–Toeplitz theorem states that an everywhere defined symmetric operator on a Hilbert space is bounded. 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, 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. 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. 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. Observables 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. 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) 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).".
- Hellinger–Toeplitz_theorem wikiPageID "592897".
- Hellinger–Toeplitz_theorem wikiPageRevisionID "551243853".
- Hellinger–Toeplitz_theorem subject Category:Functional_analysis.
- Hellinger–Toeplitz_theorem subject Category:Theorems_in_functional_analysis.
- Hellinger–Toeplitz_theorem comment "In functional analysis, a branch of mathematics, the Hellinger–Toeplitz theorem states that an everywhere defined symmetric operator on a Hilbert space is bounded. 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, 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.".
- Hellinger–Toeplitz_theorem label "Hellinger–Toeplitz theorem".
- Hellinger–Toeplitz_theorem label "Satz von Hellinger-Toeplitz".
- Hellinger–Toeplitz_theorem label "Теорема Хеллингера — Тёплица".
- Hellinger–Toeplitz_theorem label "黑林格－特普利茨定理".
- Hellinger–Toeplitz_theorem sameAs Hellinger%E2%80%93Toeplitz_theorem.
- Hellinger–Toeplitz_theorem sameAs Satz_von_Hellinger-Toeplitz.
- Hellinger–Toeplitz_theorem sameAs Q1472120.
- Hellinger–Toeplitz_theorem sameAs Q1472120.
- Hellinger–Toeplitz_theorem wasDerivedFrom Hellinger–Toeplitz_theorem?oldid=551243853.