Banach–Steinhaus theorem

E394468

result in functional analysis theorem in functional analysis uniform boundedness principle

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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All labels observed (6)

Label	Occurrences
Banach–Steinhaus theorem canonical	5
Banach–Steinhaus Theorem	1
Uniform Boundedness Principle	1
principle of uniform boundedness in Banach spaces	1
uniform boundedness principle	1
uniform boundedness theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T3884667 — 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: Banach–Steinhaus theorem
Context triple: [Hugo Steinhaus, notableWork, Banach–Steinhaus 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. Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
C. Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
D. Schauder fixed-point theorem
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
E. Banach fixed-point theorem
The Banach fixed-point theorem is a fundamental result in metric space theory that guarantees the existence and uniqueness of a fixed point for any contraction mapping and provides a method for finding it via iterative approximation.
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: Banach–Steinhaus theorem
Target entity description: 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.

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. Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
C. Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
D. Schauder fixed-point theorem
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
E. Banach fixed-point theorem
The Banach fixed-point theorem is a fundamental result in metric space theory that guarantees the existence and uniqueness of a fixed point for any contraction mapping and provides a method for finding it via iterative approximation.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	result in functional analysis ⓘ theorem in functional analysis ⓘ uniform boundedness principle ⓘ
alsoKnownAs	uniform boundedness principle ⓘ linked to: Banach–Steinhaus theorem uniform boundedness theorem ⓘ linked to: Banach–Steinhaus theorem
appliesTo	Banach spaces ⓘ normed vector spaces ⓘ
assumes	each operator in the family is continuous ⓘ each operator in the family is linear ⓘ pointwise boundedness on the Banach space ⓘ
centralConcept	Baire category ⓘ pointwise boundedness ⓘ uniform boundedness of operator norms ⓘ
characterizes	uniform boundedness of families of operators ⓘ
concerns	families of continuous linear operators ⓘ pointwise bounded families of operators ⓘ
context	linear operators between Banach spaces ⓘ locally convex topological vector spaces ⓘ
domain	topological vector spaces ⓘ
ensures	continuity of limit of uniformly bounded sequence of continuous linear operators ⓘ local boundedness of pointwise bounded families of continuous linear maps ⓘ
field	functional analysis ⓘ
hasConsequence	pathological examples in incomplete spaces ⓘ pointwise convergence of bounded operators implies bounded limit operator ⓘ
hasVariant	version for Fréchet spaces ⓘ version for barrelled spaces ⓘ
historicalPeriod	early 20th century ⓘ
holdsIn	complete normed spaces ⓘ
implies	boundedness of suprema of operator norms ⓘ equicontinuity of the family of operators on bounded sets ⓘ existence of a uniform bound on operator norms ⓘ
importance	fundamental tool in modern analysis ⓘ
involves	bounded linear operators ⓘ continuous linear functionals ⓘ
isEquivalentTo	principle of uniform boundedness in Banach spaces ⓘ linked to: Banach–Steinhaus theorem
isPartOf	Banach space theory ⓘ linked to: Banach spaces
namedAfter	Hugo Steinhaus ⓘ Stefan Banach ⓘ
proofTechnique	Baire category argument ⓘ
relatedTo	Hahn–Banach theorem ⓘ closed graph theorem ⓘ linked to: Closed Graph Theorem open mapping theorem ⓘ
requires	Baire category theorem ⓘ
usedFor	establishing regularity of limit operators ⓘ proving existence of unbounded operators when pointwise boundedness fails ⓘ studying convergence of sequences of operators ⓘ studying series of operators ⓘ

How these facts were elicited

Referenced by (10)

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

Hugo Steinhaus → notableWork → Banach–Steinhaus theorem ⓘ

Banach space → hasConcept → Banach–Steinhaus theorem ⓘ

subject linked to: Banach spaces

Stefan Banach → notableWork → Banach–Steinhaus theorem ⓘ

Stefan Banach → eponymOf → Banach–Steinhaus theorem ⓘ

Functional analysis → fieldOfStudy → Banach–Steinhaus theorem ⓘ

subject linked to: "Functional Analysis"

Banach–Steinhaus theorem → alsoKnownAs → uniform boundedness theorem ⓘ