Banach limit
E421066
A Banach limit is a linear functional on the space of bounded sequences that extends the usual limit and assigns generalized “limits” to sequences that may not converge in the classical sense.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Banach limit canonical | 1 |
| Summability (Mathematics) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4219685 — 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 limit Context triple: [Stefan Banach, eponymOf, Banach limit]
-
A.
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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B.
Banach spaces
Banach spaces are complete normed vector spaces that provide a fundamental framework for functional analysis and the study of infinite-dimensional linear phenomena.
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C.
Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
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D.
Banach algebra
A Banach algebra is a complete normed vector space equipped with a compatible associative algebra multiplication, allowing analysis and algebra to be combined in a single structure.
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E.
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.
- 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 limit Target entity description: A Banach limit is a linear functional on the space of bounded sequences that extends the usual limit and assigns generalized “limits” to sequences that may not converge in the classical sense.
-
A.
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.
-
B.
Banach spaces
Banach spaces are complete normed vector spaces that provide a fundamental framework for functional analysis and the study of infinite-dimensional linear phenomena.
-
C.
Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
-
D.
Banach algebra
A Banach algebra is a complete normed vector space equipped with a compatible associative algebra multiplication, allowing analysis and algebra to be combined in a single structure.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
generalized limit
ⓘ
linear functional ⓘ mathematical concept ⓘ object in functional analysis ⓘ |
| appliesTo | bounded but divergent sequences like (-1)^n ⓘ |
| boundednessProperty | |L(x)|≤sup_n|x_n| for all bounded sequences x ⓘ |
| category | non-normal linear functional on ℓ∞ ⓘ |
| codomain |
ℂ
ⓘ
ℝ ⓘ |
| coincidesWith | usual limit on convergent sequences ⓘ |
| definedOn |
bounded complex sequences
ⓘ
bounded real sequences ⓘ |
| domain |
space of bounded sequences
ⓘ
ℓ∞ ⓘ |
| existenceDependsOn |
Hahn–Banach theorem
ⓘ
axiom of choice ⓘ |
| extends | usual limit of convergent sequences ⓘ |
| generalizes | limit of bounded sequences ⓘ |
| introducedBy | Stefan Banach ⓘ |
| introducedInField | functional analysis ⓘ |
| invarianceProperty | translation invariance on index set ℕ ⓘ |
| invariantUnder | finite permutations of coordinates of a sequence (for some constructions) ⓘ |
| linearityProperty |
L(x+y)=L(x)+L(y)
ⓘ
L(αx)=αL(x) ⓘ |
| majorizationProperty | liminf x_n ≤ L(x) ≤ limsup x_n ⓘ |
| nonConstructiveProperty | no explicit formula known for a specific Banach limit ⓘ |
| nonUniquenessProperty | there exist many distinct Banach limits ⓘ |
| normProperty | ‖L‖=1 ⓘ |
| notCountablyAdditive | not given by a countably additive measure on ℕ ⓘ |
| positivityProperty | if x_n≥0 for all n then L(x)≥0 ⓘ |
| relatedConcept |
Cesàro summation
ⓘ
Følner sequence ⓘ invariant mean ⓘ ultrafilter limit ⓘ |
| requires | choice of extension of the usual limit functional ⓘ |
| shiftInvariance | L((x_{n+1}))=L((x_n)) ⓘ |
| specialCaseExample | for convergent x_n, L(x)=lim x_n ⓘ |
| takesValueOn | bounded sequence (x_n) ⓘ |
| topologicalProperty | element of the dual space (ℓ∞)∗ ⓘ |
| usedFor |
assigning generalized limits to nonconvergent bounded sequences
ⓘ
ergodic theory ⓘ extensions of the Cesàro mean ⓘ invariant means on ℤ ⓘ studying summability of sequences ⓘ |
| usedInProofOf | mean ergodic theorems in some formulations ⓘ |
| valueConstraintOnAlternatingSequence | for x_n=(-1)^n, any Banach limit satisfies -1≤L(x)≤1 ⓘ |
| valueNotation | L(x) or L((x_n)) ⓘ |
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: Banach limit Description of subject: A Banach limit is a linear functional on the space of bounded sequences that extends the usual limit and assigns generalized “limits” to sequences that may not converge in the classical sense.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Banach limit