Montel space
E884920
A Montel space is a type of locally convex topological vector space in which every closed and bounded set is compact, implying strong convergence and compactness properties useful in functional analysis and distribution theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Montel space canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10772851 — 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: Montel space Context triple: [Produits tensoriels topologiques et espaces nucléaires, relatedConcept, Montel space]
-
A.
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.
-
B.
Schwartz–Bruhat space
The Schwartz–Bruhat space is a function space of rapidly decreasing smooth (or locally constant with compact support, in the non-Archimedean case) test functions on a locally compact abelian group, fundamental in harmonic analysis and number theory.
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C.
Orlicz spaces
Orlicz spaces are a class of function spaces that generalize Lebesgue spaces by measuring integrability via convex Orlicz functions rather than fixed power exponents.
-
D.
Lebesgue spaces
Lebesgue spaces are function spaces, denoted \(L^p\), that consist of measurable functions whose absolute values raised to the \(p\)-th power are integrable, forming a fundamental framework in modern analysis and probability theory.
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E.
Minkowski functional
The Minkowski functional is a mathematical tool in functional analysis that assigns a nonnegative real number to each vector in a vector space based on its position relative to a given convex, balanced, absorbing set, generalizing the notion of a norm.
- 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: Montel space Target entity description: A Montel space is a type of locally convex topological vector space in which every closed and bounded set is compact, implying strong convergence and compactness properties useful in functional analysis and distribution theory.
-
A.
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.
-
B.
Schwartz–Bruhat space
The Schwartz–Bruhat space is a function space of rapidly decreasing smooth (or locally constant with compact support, in the non-Archimedean case) test functions on a locally compact abelian group, fundamental in harmonic analysis and number theory.
-
C.
Orlicz spaces
Orlicz spaces are a class of function spaces that generalize Lebesgue spaces by measuring integrability via convex Orlicz functions rather than fixed power exponents.
-
D.
Lebesgue spaces
Lebesgue spaces are function spaces, denoted \(L^p\), that consist of measurable functions whose absolute values raised to the \(p\)-th power are integrable, forming a fundamental framework in modern analysis and probability theory.
-
E.
Minkowski functional
The Minkowski functional is a mathematical tool in functional analysis that assigns a nonnegative real number to each vector in a vector space based on its position relative to a given convex, balanced, absorbing set, generalizing the notion of a norm.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
locally convex space class
ⓘ
topological vector space property ⓘ |
| belongsToClass |
barrelled locally convex spaces
ⓘ
semi-reflexive locally convex spaces ⓘ spaces where every closed bounded set is compact ⓘ |
| characterizedBy |
every bounded sequence has a convergent subsequence in the strong topology (for Fréchet–Montel spaces)
ⓘ
every closed bounded set is compact in the given locally convex topology ⓘ |
| definedOn | locally convex topological vector space ⓘ |
| fieldOfStudy |
distribution theory
ⓘ
functional analysis ⓘ topological vector spaces ⓘ |
| generalizes | Montel theorem for families of holomorphic functions NERFINISHED ⓘ |
| hasConsequence |
bounded sets are relatively compact in every compatible locally convex topology
ⓘ
various versions of the Banach–Alaoglu theorem simplify ⓘ weak and strong boundedness coincide for many purposes ⓘ |
| hasDefiningProperty | every closed and bounded subset is compact ⓘ |
| hasExample |
Fréchet space of holomorphic functions on a domain in ℂ^n
ⓘ
Schwartz space S(ℝ^n) NERFINISHED ⓘ certain spaces of distributions with appropriate topologies ⓘ space of real-analytic functions on an open set ⓘ space of test functions C_c^∞(Ω) ⓘ |
| hasHistoricalContext | introduced in connection with normal families of holomorphic functions ⓘ |
| hasTypicalTopology | locally convex topology defined by a family of seminorms ⓘ |
| impliesProperty |
barrelled space
ⓘ
bornological space ⓘ every bounded set is precompact ⓘ every bounded subset is relatively compact ⓘ every closed bounded set is complete ⓘ every sequence in a bounded set has a convergent subsequence (for metrizable cases) ⓘ every weakly convergent sequence is strongly convergent (in many classical examples) ⓘ reflexive space (in the locally convex sense) ⓘ semi-reflexive space ⓘ |
| isStrongerThan |
barrelled space
ⓘ
bornological space ⓘ reflexive locally convex space ⓘ |
| isWeakerThan | nuclear space (in many standard hierarchies) ⓘ |
| namedAfter | Paul Montel NERFINISHED ⓘ |
| oftenAssumedToBe |
Hausdorff
NERFINISHED
ⓘ
complete ⓘ |
| relatedConcept |
DF-space (dual of a Fréchet space)
ⓘ
Fréchet–Montel space NERFINISHED ⓘ nuclear space ⓘ |
| topologicalDualProperty |
the strong dual of a Montel space is Montel
ⓘ
the strong dual of a Montel space is barrelled ⓘ the strong dual of a Montel space is reflexive ⓘ |
| usedIn |
compactness arguments in functional analysis
ⓘ
partial differential equations ⓘ theory of distributions ⓘ theory of holomorphic functions ⓘ |
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: Montel space Description of subject: A Montel space is a type of locally convex topological vector space in which every closed and bounded set is compact, implying strong convergence and compactness properties useful in functional analysis and distribution theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.