Montel theorem
E259771
Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Montel theorem canonical | 2 |
| Arzelà–Ascoli theorem for holomorphic functions | 1 |
| Montel's theorem for meromorphic functions | 1 |
| Montel's theorem for subharmonic functions | 1 |
| Picard's first theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2364536 — 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: Montel theorem Context triple: [Riemann mapping theorem, proofMethod, Montel theorem]
-
A.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
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B.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
-
C.
Weierstrass factorization theorem
The Weierstrass factorization theorem is a fundamental result in complex analysis that expresses any entire function as an infinite product determined by its zeros, generalizing the factorization of polynomials.
-
D.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
E.
Cauchy integral theorem
The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
- 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: Montel theorem Target entity description: Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
-
A.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
-
B.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
-
C.
Weierstrass factorization theorem
The Weierstrass factorization theorem is a fundamental result in complex analysis that expresses any entire function as an infinite product determined by its zeros, generalizing the factorization of polynomials.
-
D.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
E.
Cauchy integral theorem
The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
result in normal family theory
ⓘ
theorem in complex analysis ⓘ |
| appliesTo | families of holomorphic functions ⓘ |
| assumes |
complex-valued functions
ⓘ
holomorphy on a common domain ⓘ |
| conclusion |
every sequence in the family has a subsequence that converges uniformly on compact subsets
ⓘ
limit of a convergent subsequence is holomorphic on the domain ⓘ the family is a normal family ⓘ |
| condition |
family is defined on a common domain
ⓘ
family is uniformly bounded on every compact subset of the domain ⓘ functions are holomorphic on the domain ⓘ |
| consequence |
compactness criteria for families of holomorphic functions
ⓘ
existence of convergent subsequences of holomorphic functions ⓘ |
| dealsWith |
holomorphic functions
ⓘ
normal families ⓘ uniform convergence on compact sets ⓘ |
| domainType |
domain in the complex plane
ⓘ
open subset of the complex plane ⓘ |
| field |
complex analysis
ⓘ
function theory ⓘ |
| generalizationOf |
Montel theorem
self-linksurface differs
ⓘ
surface form:
Arzelà–Ascoli theorem for holomorphic functions
|
| hasVariant |
Montel theorem
self-linksurface differs
ⓘ
surface form:
Montel's theorem for meromorphic functions
Montel theorem self-linksurface differs ⓘ
surface form:
Montel's theorem for subharmonic functions
|
| historicalPeriod | early 20th century ⓘ |
| implies |
precompactness of the family in the compact-open topology
ⓘ
relative compactness in the topology of uniform convergence on compact sets ⓘ |
| mathematicsSubjectClassification | 30-XX Complex functions ⓘ |
| namedAfter | Paul Montel ⓘ |
| relatedTo |
Hurwitz theorem
ⓘ
surface form:
Hurwitz's theorem
Picard theorem ⓘ
surface form:
Picard's theorem
Riemann mapping theorem ⓘ Vitali convergence theorem ⓘ normal family ⓘ |
| topologyUsed |
compact-open topology
ⓘ
topology of uniform convergence on compact sets ⓘ |
| typicalFormulation | A family of holomorphic functions on a domain that is uniformly bounded on every compact subset is normal ⓘ |
| usedIn |
complex dynamics
ⓘ
dynamics of rational maps ⓘ iteration theory of holomorphic functions ⓘ proofs of Picard-type theorems ⓘ value distribution theory ⓘ |
| usesConcept |
Arzelà–Ascoli theorem
ⓘ
Cauchy integral formula ⓘ
surface form:
Cauchy estimates
compact subsets of complex domains ⓘ equicontinuity ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Montel theorem Description of subject: Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Picard's first theorem
subject surface form:
Montel's theorem
this entity surface form:
Arzelà–Ascoli theorem for holomorphic functions
subject surface form:
Montel's theorem
this entity surface form:
Montel's theorem for meromorphic functions
subject surface form:
Montel's theorem
this entity surface form:
Montel's theorem for subharmonic functions