Riesz–Fischer theorem
E384563
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Riesz–Fischer theorem canonical | 5 |
How this entity was disambiguated
This entity first appeared as the object of triple T3748400 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Riesz–Fischer theorem Context triple: [Erhard Schmidt, notableWork, Riesz–Fischer theorem]
-
A.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
-
B.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
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C.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
-
D.
Khinchin's representation theorem
Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
-
E.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Riesz–Fischer theorem Target entity description: The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
-
A.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
-
B.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
C.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
-
D.
Khinchin's representation theorem
Khinchin's representation theorem is a result in probability theory that characterizes stationary stochastic processes by representing them in terms of simpler, more fundamental random components.
-
E.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in functional analysis ⓘ |
| appliesTo |
L^2 of a measure space
ⓘ
L^2[0,1] ⓘ L^2[0,2π] ⓘ separable Hilbert spaces ⓘ |
| assumes |
complete inner product space structure on L^2
ⓘ
countable orthonormal system ⓘ |
| category |
theorem about Hilbert spaces
ⓘ
theorem about L^2 convergence ⓘ |
| conclusion |
image of l^2 under the expansion map is closed in L^2
ⓘ
mapping from l^2 to L^2 given by orthonormal expansion is linear and isometric ⓘ series with l^2 coefficients converges in L^2 to a unique function ⓘ |
| field |
Fourier analysis
ⓘ
functional analysis ⓘ measure theory ⓘ |
| historicalContext | proved in the early 20th century ⓘ |
| implies |
L^2 is a Hilbert space
ⓘ
Parseval identity for orthonormal systems in L^2 ⓘ completeness of L^2 spaces ⓘ existence of orthogonal expansions in L^2 ⓘ |
| importance |
fundamental for the abstract theory of Hilbert spaces
ⓘ
provides rigorous basis for Fourier analysis in L^2 ⓘ |
| mainStatement |
establishes an isometric isomorphism between l^2 and L^2 spaces associated with orthonormal systems
ⓘ
every square-summable sequence of coefficients defines an L^2 function via an orthonormal basis ⓘ the L^2 norm of a function equals the l^2 norm of its coefficients with respect to an orthonormal basis ⓘ |
| namedAfter |
Ernst Sigismund Fischer
ⓘ
Frigyes Riesz ⓘ |
| relatedTo |
Bessel inequality
ⓘ
Parseval theorem ⓘ
linked to:
Parseval's theorem
Plancherel theorem ⓘ completeness of orthonormal systems ⓘ |
| relatesConcept |
Fourier coefficients
ⓘ
Fourier series ⓘ Hilbert space ⓘ
linked to:
Hilbert spaces
L^2 space ⓘ completeness ⓘ inner product space ⓘ isometric isomorphism ⓘ orthonormal basis ⓘ orthonormal system ⓘ square-integrable functions ⓘ square-summable sequences ⓘ |
| usedFor |
construction of L^2 spaces
ⓘ
foundation of modern Hilbert space theory ⓘ justification of Fourier series convergence in L^2 ⓘ representation of elements of L^2 by orthonormal expansions ⓘ spectral theory of self-adjoint operators ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Riesz–Fischer theorem Description of subject: The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.