Riesz representation theorem

E621089

mathematical theorem result in Hilbert space theory

The Riesz representation theorem is a fundamental result in functional analysis that characterizes continuous linear functionals on Hilbert spaces as inner products with a unique vector in the space.

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Observed surface forms (1)

Surface form	Occurrences
Riesz–Markov–Kakutani representation theorem	1

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in Hilbert space theory ⓘ
appearsIn	textbooks on Hilbert space theory ⓘ
appliesTo	Hilbert space NERFINISHED ⓘ non-separable Hilbert spaces ⓘ separable Hilbert spaces ⓘ spaces of square-integrable functions ⓘ
characterizes	continuous linear functionals on Hilbert spaces ⓘ
classification	representation theorem NERFINISHED ⓘ
codomainObject	vector in the Hilbert space ⓘ
domainObject	continuous linear functional ⓘ
equivalentTo	identifying each continuous linear functional with an inner product against a fixed vector ⓘ
field	functional analysis ⓘ operator theory ⓘ
guarantees	existence of a representing vector for each continuous linear functional on a Hilbert space ⓘ uniqueness of the representing vector ⓘ
hasVersion	Riesz representation theorem for Hilbert spaces NERFINISHED ⓘ Riesz representation theorem for measures NERFINISHED ⓘ
holdsIn	complex Hilbert spaces ⓘ real Hilbert spaces ⓘ
implies	every Hilbert space is reflexive ⓘ norm of the functional equals the norm of the representing vector ⓘ the continuous dual of a Hilbert space is isometrically isomorphic to the Hilbert space itself ⓘ
involvesConcept	Hilbert space NERFINISHED ⓘ adjoint operator ⓘ continuous linear functional ⓘ duality ⓘ inner product ⓘ norm ⓘ orthogonality ⓘ
namedAfter	Frigyes Riesz NERFINISHED ⓘ
provides	an isometric isomorphism between a Hilbert space and its dual ⓘ
relatedTo	Hahn–Banach theorem NERFINISHED ⓘ Lax–Milgram theorem NERFINISHED ⓘ Riesz–Fréchet representation theorem NERFINISHED ⓘ Riesz–Markov–Kakutani representation theorem NERFINISHED ⓘ
relates	a Hilbert space to its continuous dual space ⓘ
requires	completeness of the inner product space ⓘ continuity of the linear functional ⓘ
specialCaseOf	duality theory in Banach spaces ⓘ
statesThat	every continuous linear functional on a Hilbert space can be represented as an inner product with a unique vector in that space ⓘ
topicOf	many graduate-level functional analysis courses ⓘ
usedFor	Lax–Milgram theorem applications NERFINISHED ⓘ defining adjoint operators on Hilbert spaces ⓘ identifying the dual of a Hilbert space ⓘ spectral theory of self-adjoint operators ⓘ variational formulations of boundary value problems ⓘ weak formulations in partial differential equations ⓘ

Referenced by (8)

Full triples — surface form annotated when it differs from this entity's canonical label.

linear algebra → hasKeyTheorem → Riesz representation theorem ⓘ

Frigyes Riesz → knownFor → Riesz representation theorem ⓘ

this entity surface form: Riesz–Markov–Kakutani representation theorem

Gelfand representation of commutative C*-algebras → relatesTo → Riesz representation theorem ⓘ

"Functional Analysis" → fieldOfStudy → Riesz representation theorem ⓘ

subject surface form: Functional analysis

Banach–Alaoglu theorem → relatedTo → Riesz representation theorem ⓘ

Hahn–Banach theorem → relatedTo → Riesz representation theorem ⓘ

Banach–Stone theorem → relatedTo → Riesz representation theorem ⓘ