Closed Graph Theorem

E412931

mathematical theorem theorem in functional analysis

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.

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All labels observed (3)

Label	Occurrences
closed graph theorem	4
Closed graph theorem	2
Closed Graph Theorem canonical	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]

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
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 ⓘ linked to: Banach–Steinhaus theorem Bounded Inverse Theorem ⓘ linked to: Banach inverse mapping theorem Open Mapping Theorem ⓘ linked to: open mapping theorem Uniform Boundedness Principle ⓘ linked to: Banach–Steinhaus theorem
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	spectral theory of linear operators ⓘ linked to: functional analysis 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

Referenced by (7)

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

Banach space → hasConcept → Closed Graph Theorem ⓘ

subject linked to: Banach spaces

Banach inverse mapping theorem → relatedTo → closed graph theorem ⓘ