Hilbert spaces
E2126
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Hilbert space | 2 |
| Fock space | 1 |
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
complete metric space
ⓘ
inner product space ⓘ mathematical structure ⓘ vector space ⓘ |
| hasConcept |
bounded linear operator
ⓘ
closed subspace ⓘ compact operator ⓘ orthogonal projection ⓘ orthonormal basis ⓘ self-adjoint operator ⓘ unitary operator ⓘ |
| hasDefinition | a complete inner product space ⓘ |
| hasExample |
Hardy space H^2 on the unit disk
ⓘ
Sobolev space H^1 with appropriate inner product ⓘ finite-dimensional Euclidean space R^n with standard inner product ⓘ finite-dimensional complex space C^n with standard inner product ⓘ function space L^2 of square-integrable functions ⓘ sequence space l^2 of square-summable sequences ⓘ |
| hasOperation |
inner product conjugate symmetric in complex case
ⓘ
inner product linear in first argument (in complex case, conjugate-linear in second) ⓘ inner product positive definite ⓘ inner product symmetric in real case ⓘ |
| hasProperty |
Bessel inequality holds for orthonormal systems
ⓘ
Cauchy sequences converge in the norm ⓘ Parseval identity holds for complete orthonormal systems ⓘ Pythagorean theorem generalizes to orthogonal vectors ⓘ Riesz representation theorem holds for continuous linear functionals ⓘ completeness with respect to the norm induced by the inner product ⓘ every closed subspace has an orthogonal complement ⓘ every vector has a unique decomposition into components in a closed subspace and its orthogonal complement ⓘ is a Banach space ⓘ isometries preserve inner product and norm in the Hilbert space ⓘ parallelogram law holds for the norm ⓘ projection theorem holds for closed convex subsets ⓘ separable if it has a countable dense subset ⓘ |
| hasStructure |
inner product
ⓘ
scalar multiplication ⓘ vector addition ⓘ |
| induces | norm via the inner product ⓘ |
| isA | Banach space with norm from an inner product ⓘ |
| mayBe |
complex
ⓘ
real ⓘ |
| namedAfter | David Hilbert ⓘ |
| usedIn |
control theory
ⓘ
functional analysis ⓘ harmonic analysis ⓘ partial differential equations ⓘ quantum field theory ⓘ quantum mechanics ⓘ signal processing ⓘ |
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hilbert space
subject surface form:
von Neumann algebra
this entity surface form:
Hilbert space
this entity surface form:
Fock space