Dini's theorem
E877685
Dini's theorem is a result in real analysis that gives conditions under which a monotone sequence of continuous functions converging pointwise on a compact space actually converges uniformly.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Dini's theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10660048 — 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: Dini's theorem Context triple: [Ulisse Dini, knownFor, Dini's theorem]
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A.
Gale–Nikaidō–Debreu theorem
The Gale–Nikaidō–Debreu theorem is a fundamental result in mathematical economics that provides conditions ensuring the existence (and sometimes uniqueness) of equilibrium in certain nonlinear and general equilibrium models.
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B.
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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C.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
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D.
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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E.
Krein–Milman theorem
The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
- 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: Dini's theorem Target entity description: Dini's theorem is a result in real analysis that gives conditions under which a monotone sequence of continuous functions converging pointwise on a compact space actually converges uniformly.
-
A.
Gale–Nikaidō–Debreu theorem
The Gale–Nikaidō–Debreu theorem is a fundamental result in mathematical economics that provides conditions ensuring the existence (and sometimes uniqueness) of equilibrium in certain nonlinear and general equilibrium models.
-
B.
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.
-
C.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
-
D.
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.
-
E.
Krein–Milman theorem
The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appliesTo |
compact metric spaces
ⓘ
compact subsets of ℝ ⓘ compact topological spaces ⓘ |
| assumes |
compactness of the domain
ⓘ
monotone convergence in n for each point ⓘ pointwise convergence to a continuous limit function ⓘ sequence of continuous real-valued functions ⓘ |
| concerns |
continuous functions
ⓘ
monotone sequence of functions ⓘ pointwise convergence ⓘ uniform convergence ⓘ |
| concludes | uniform convergence of the sequence to the limit function ⓘ |
| failsIf |
the domain is not compact
ⓘ
the limit function is not continuous ⓘ the sequence is not monotone ⓘ |
| field |
mathematical analysis
ⓘ
real analysis ⓘ |
| hasVersion |
decreasing sequence version
ⓘ
increasing sequence version ⓘ topological space formulation ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| holdsFor |
complex-valued functions
ⓘ
continuous functions on compact Hausdorff spaces ⓘ real-valued functions ⓘ |
| implies |
interchange of limit and supremum under its hypotheses
ⓘ
limit function is continuous ⓘ |
| namedAfter | Ulisse Dini ⓘ |
| namedEntityType | mathematical result ⓘ |
| relatedTo |
Arzelà–Ascoli theorem
ⓘ
Lebesgue dominated convergence theorem ⓘ
linked to:
dominated convergence theorem
Weierstrass M-test ⓘ monotone convergence theorem ⓘ |
| requires |
continuity of the limit function
ⓘ
monotonicity of the sequence at each point ⓘ |
| typicalStatement | If (f_n) is a monotone sequence of continuous real-valued functions on a compact space K converging pointwise to a continuous function f, then (f_n) converges uniformly to f on K. ⓘ |
| usedIn |
Fourier analysis
ⓘ
approximation theory ⓘ functional analysis ⓘ potential theory ⓘ proofs of the Weierstrass approximation theorem ⓘ |
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: Dini's theorem Description of subject: Dini's theorem is a result in real analysis that gives conditions under which a monotone sequence of continuous functions converging pointwise on a compact space actually converges uniformly.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.