Gelfand triples (rigged Hilbert spaces)
E270387
concept in functional analysis
concept in quantum mechanics
mathematical structure
rigged Hilbert space
Gelfand triples (rigged Hilbert spaces) are a mathematical framework that extends Hilbert spaces to rigorously handle generalized eigenvectors and distributions, particularly in quantum mechanics and functional analysis.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Gelfand triples (rigged Hilbert spaces) canonical | 1 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
concept in functional analysis
ⓘ
concept in quantum mechanics ⓘ mathematical structure ⓘ rigged Hilbert space ⓘ |
| alsoKnownAs | rigged Hilbert space ⓘ |
| appliesTo |
Hamiltonian operators in quantum mechanics
ⓘ
momentum operator in quantum mechanics ⓘ position operator in quantum mechanics ⓘ unbounded operators on Hilbert spaces ⓘ |
| component |
Hilbert space H
ⓘ
dual space Φ′ ⓘ test function space Φ ⓘ |
| enables |
definition of generalized eigenvectors as continuous antilinear functionals on Φ
ⓘ
extension of the spectral theorem to continuous spectrum ⓘ rigorous treatment of scattering states ⓘ use of distribution-valued eigenfunctions ⓘ |
| example |
Schwartz space
ⓘ
surface form:
Schwartz space S(ℝⁿ) ⊂ L²(ℝⁿ) ⊂ S′(ℝⁿ)
space of smooth compactly supported functions C_c^∞(Ω) ⊂ L²(Ω) ⊂ distributions D′(Ω) ⓘ |
| field |
operator theory
ⓘ
theory of topological vector spaces ⓘ |
| formalDefinition | a triplet of spaces Φ ⊂ H ⊂ Φ′ where H is a Hilbert space, Φ is a dense subspace of H with a finer topology, and Φ′ is the continuous dual of Φ ⓘ |
| generalizes | Hilbert space framework for quantum mechanics ⓘ |
| hasDual | continuous dual space Φ′ of Φ ⓘ |
| hasTopology | locally convex topology on Φ ⓘ |
| historicalContext | developed in the mid-20th century ⓘ |
| namedAfter | Israel Gelfand ⓘ |
| property |
H is continuously embedded in Φ′
ⓘ
embedding Φ → H is continuous ⓘ Φ carries a locally convex topology stronger than the Hilbert space topology induced from H ⓘ Φ is densely embedded in H ⓘ |
| purpose |
to describe continuous spectrum eigenstates
ⓘ
to extend the spectral theory of unbounded operators ⓘ to handle non-normalizable states in quantum mechanics ⓘ to incorporate distributions into Hilbert space methods ⓘ to provide a framework for Dirac bra–ket formalism ⓘ to rigorously treat generalized eigenvectors ⓘ |
| relatedTo |
Dirac delta function
ⓘ
surface form:
Dirac delta distribution
Schwartz space ⓘ distribution (generalized function) ⓘ generalized eigenvector ⓘ rigorous formulation of Dirac notation ⓘ self-adjoint operator ⓘ spectral decomposition ⓘ tempered distributions ⓘ |
| usedIn |
distribution theory
ⓘ
functional analysis ⓘ mathematical physics ⓘ quantum mechanics ⓘ spectral theory ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.