Closed Graph Theorem
E412931
The Closed Graph Theorem is a fundamental result in functional analysis stating that a linear operator between Banach spaces is bounded (and hence continuous) if its graph is closed in the product space.
All labels observed (3)
| Label | Occurrences |
|---|---|
| closed graph theorem | 3 |
| Closed Graph Theorem canonical | 1 |
| Closed graph theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4092297 — 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: Closed Graph Theorem Context triple: [Banach space, hasConcept, Closed Graph Theorem]
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A.
Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
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B.
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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C.
Tarski’s fixed point theorem
Tarski’s fixed point theorem is a fundamental result in order theory and lattice theory that guarantees the existence of fixed points for monotone functions on complete lattices, with wide applications in logic, computer science, and economics.
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D.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
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E.
Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
- 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: Closed Graph Theorem Target entity description: The Closed Graph Theorem is a fundamental result in functional analysis stating that a linear operator between Banach spaces is bounded (and hence continuous) if its graph is closed in the product space.
-
A.
Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
-
B.
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.
-
C.
Tarski’s fixed point theorem
Tarski’s fixed point theorem is a fundamental result in order theory and lattice theory that guarantees the existence of fixed points for monotone functions on complete lattices, with wide applications in logic, computer science, and economics.
-
D.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
-
E.
Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in functional analysis ⓘ |
| appliesTo |
Banach spaces
ⓘ
linear operators ⓘ |
| assumes | operator is defined on all of the Banach space ⓘ |
| bidirectional | closed graph if and only if bounded for linear maps between Banach spaces ⓘ |
| characterizes | continuity of linear operators between Banach spaces ⓘ |
| codomainCondition | codomain is a Banach space ⓘ |
| conclusion |
operator is bounded
ⓘ
operator is continuous ⓘ |
| consequence | any everywhere-defined closed linear operator between Banach spaces is continuous ⓘ |
| context | topological vector spaces ⓘ |
| distinguishes | closed operators from merely closable operators ⓘ |
| domainCondition | domain is a Banach space ⓘ |
| equivalence | A linear operator between Banach spaces is continuous if and only if its graph is closed in the product space ⓘ |
| failsIn | general normed spaces that are not complete ⓘ |
| field | functional analysis ⓘ |
| generalizationOf | results about continuity of linear maps in finite-dimensional spaces ⓘ |
| hasVariant |
versions for Fréchet spaces
ⓘ
versions for locally convex spaces under additional hypotheses ⓘ |
| holdsIn | normed linear spaces that are complete ⓘ |
| hypothesis |
graph of the operator is closed in the product space
ⓘ
operator is linear ⓘ |
| implies |
closed densely defined unbounded operators cannot be everywhere-defined on a Banach space
ⓘ
closed graph implies bounded operator ⓘ closed graph implies continuous operator ⓘ graph of a bounded linear operator between Banach spaces is closed ⓘ |
| logicalForm |
T is bounded implies graph(T) is closed in X×Y
ⓘ
if graph(T) is closed in X×Y then T is bounded ⓘ |
| mathematicalArea | analysis ⓘ |
| relatedTo |
Banach–Steinhaus theorem
ⓘ
surface form:
Banach–Steinhaus Theorem
Banach inverse mapping theorem ⓘ
surface form:
Bounded Inverse Theorem
open mapping theorem ⓘ
surface form:
Open Mapping Theorem
Banach–Steinhaus theorem ⓘ
surface form:
Uniform Boundedness Principle
|
| requires |
completeness of codomain space
ⓘ
completeness of domain space ⓘ product of Banach spaces is a Banach space ⓘ |
| statement | A linear operator between Banach spaces is bounded if its graph is closed in the product space ⓘ |
| typicalFormulation | If X and Y are Banach spaces and T:X→Y is linear with closed graph in X×Y, then T is bounded ⓘ |
| usedFor |
proving continuity of linear operators
ⓘ
showing unbounded operators cannot have closed graphs unless domain is restricted ⓘ |
| usedIn |
functional analysis
ⓘ
surface form:
spectral theory of linear operators
study of unbounded operators on Hilbert spaces ⓘ theory of partial differential operators ⓘ |
| usesConcept |
closed set
ⓘ
graph of an operator ⓘ product of Banach spaces ⓘ |
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: Closed Graph Theorem Description of subject: The Closed Graph Theorem is a fundamental result in functional analysis stating that a linear operator between Banach spaces is bounded (and hence continuous) if its graph is closed in the product space.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Banach space
this entity surface form:
closed graph theorem
subject surface form:
Functional analysis
this entity surface form:
Closed graph theorem
this entity surface form:
closed graph theorem
this entity surface form:
closed graph theorem