open mapping theorem

E518476

theorem in functional analysis theorem in functional analysis of Banach spaces

The open mapping theorem is a fundamental result in functional analysis stating that any surjective continuous linear operator between Banach spaces maps open sets to open sets.

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All labels observed (3)

Label	Occurrences
open mapping theorem canonical	3
Open mapping theorem	2
Open Mapping Theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T5425744 — 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: open mapping theorem
Context triple: [Banach inverse mapping theorem, implies, open mapping theorem]

A. Banach inverse mapping theorem
The Banach inverse mapping theorem is a fundamental result in functional analysis stating that a bijective bounded linear operator between Banach spaces has a bounded linear inverse.
B. 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.
C. Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
D. Montel theorem
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.
E. Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
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: open mapping theorem
Target entity description: The open mapping theorem is a fundamental result in functional analysis stating that any surjective continuous linear operator between Banach spaces maps open sets to open sets.

A. Banach inverse mapping theorem
The Banach inverse mapping theorem is a fundamental result in functional analysis stating that a bijective bounded linear operator between Banach spaces has a bounded linear inverse.
B. 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.
C. Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
D. Montel theorem
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.
E. Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	theorem in functional analysis ⓘ theorem in functional analysis of Banach spaces ⓘ
appliesTo	Banach spaces ⓘ
assumption	codomain space is complete ⓘ domain space is complete ⓘ operator is continuous ⓘ operator is linear ⓘ operator is surjective ⓘ
category	topological vector space theorem ⓘ
conclusion	a surjective bounded linear operator between Banach spaces is an open map ⓘ surjective continuous linear operators map open sets to open sets ⓘ
consequence	range of a surjective bounded linear operator between Banach spaces is automatically open in the codomain topology ⓘ
dependsOn	Baire category theorem ⓘ
doesNotHoldIf	codomain is not complete in general ⓘ domain is not complete in general ⓘ
domainCondition	continuous linear operator between Banach spaces ⓘ operator is surjective ⓘ
ensures	local surjectivity properties of bounded linear operators ⓘ
equivalentTo	bounded inverse theorem under suitable hypotheses ⓘ
field	functional analysis ⓘ
generalizationOf	open mapping results for finite-dimensional normed spaces ⓘ
generalizedBy	open mapping theorem for Fréchet spaces ⓘ open mapping theorem for barrelled spaces ⓘ
historicalContext	proved in the early development of Banach space theory in the 20th century ⓘ
holdsFor	complex Banach spaces ⓘ real Banach spaces ⓘ
implies	continuous linear surjections between Banach spaces are quotient maps ⓘ if T is bijective bounded linear operator between Banach spaces then T^{-1} is bounded ⓘ if T is surjective and bounded then T(B_X(0,1)) contains a ball around 0 in Y ⓘ images of open balls are neighborhoods of the image point ⓘ
mathematicalSubjectClassification	46Axx ⓘ 46Bxx ⓘ
proofTechnique	Baire category theorem ⓘ
relatedTo	bounded inverse theorem ⓘ closed graph theorem ⓘ
requires	norm topology on Banach spaces ⓘ
statementStyle	global property of linear operators ⓘ
typicalFormulation	If X and Y are Banach spaces and T:X→Y is bounded, linear, and surjective, then T is an open map ⓘ
usedIn	Banach space theory ⓘ linked to: Banach spaces distribution theory ⓘ partial differential equations ⓘ spectral theory of linear operators ⓘ study of solvability of linear operator equations ⓘ
usedInProofOf	inverse mapping theorem for Banach spaces ⓘ
usedToShow	closed range plus surjectivity implies quantitative bounds on inverses ⓘ solution operators for certain linear equations are continuous ⓘ

How these facts were elicited

Referenced by (6)

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

Banach inverse mapping theorem → implies → open mapping theorem ⓘ

Functional analysis → fieldOfStudy → Open mapping theorem ⓘ

subject linked to: "Functional Analysis"

linked to: open mapping theorem

Banach–Steinhaus theorem → relatedTo → open mapping theorem ⓘ

Closed Graph Theorem → relatedTo → Open Mapping Theorem ⓘ

linked to: open mapping theorem

Hahn–Banach theorem → relatedTo → open mapping theorem ⓘ

Baire category theorem → hasConsequence → Open mapping theorem ⓘ

linked to: open mapping theorem

All labels observed (3)

How this entity was disambiguated Show

Statements (46)

How these facts were elicited Show

Referenced by (6)

How this entity was disambiguated

How these facts were elicited