open mapping theorem
E518476
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| open mapping theorem canonical | 3 |
| Open Mapping Theorem | 1 |
| 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.
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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.
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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.
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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.
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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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
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
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: open mapping theorem Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Functional analysis
this entity surface form:
Open mapping theorem
this entity surface form:
Open Mapping Theorem