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 | 2 |
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's theorem ⓘ
surface form:
Parseval theorem
Plancherel theorem for locally compact abelian groups ⓘ
surface form:
Plancherel theorem
completeness of orthonormal systems ⓘ |
| relatesConcept |
Fourier coefficients
ⓘ
Fourier series ⓘ Hilbert spaces ⓘ
surface form:
Hilbert space
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
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Instruction
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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 (2)
Full triples — surface form annotated when it differs from this entity's canonical label.