Montel theorem

E259771

result in normal family theory theorem in complex analysis

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.

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All labels observed (6)

Label	Occurrences
Montel theorem canonical	5
Arzelà–Ascoli theorem for holomorphic functions	1
Montel theorem for families of 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.
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	Arzelà–Ascoli theorem for holomorphic functions ⓘ linked to: Montel theorem
hasVariant	Montel's theorem for meromorphic functions ⓘ linked to: Montel theorem Montel's theorem for subharmonic functions ⓘ linked to: Montel theorem
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's theorem ⓘ linked to: Hurwitz theorem Picard's theorem ⓘ linked to: Picard 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 estimates ⓘ linked to: Cauchy integral formula compact subsets of complex domains ⓘ equicontinuity ⓘ

How these facts were elicited

Referenced by (10)

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

Riemann mapping theorem → proofMethod → Montel theorem ⓘ

Émile Picard → notableWork → Picard's first theorem ⓘ

linked to: Montel theorem

Montel's theorem → generalizationOf → Arzelà–Ascoli theorem for holomorphic functions ⓘ

subject linked to: Montel theorem

linked to: Montel theorem

Montel's theorem → hasVariant → Montel's theorem for meromorphic functions ⓘ

subject linked to: Montel theorem

linked to: Montel theorem

Montel's theorem → hasVariant → Montel's theorem for subharmonic functions ⓘ

subject linked to: Montel theorem

linked to: Montel theorem

Picard theorem → hasProofMethod → Montel theorem ⓘ

Functions of One Complex Variable → topic → Montel theorem ⓘ

subject linked to: Functions of One Complex Variable by John B. Conway

Montel space → generalizes → Montel theorem for families of holomorphic functions ⓘ

linked to: Montel theorem

Paul Montel → notableConcept → Montel theorem ⓘ

Fatou set → relatedConcept → Montel theorem ⓘ

subject linked to: Fatou sets

All labels observed (6)

How this entity was disambiguated Show

Statements (45)

How these facts were elicited Show

Referenced by (10)

How this entity was disambiguated

How these facts were elicited