Schwartz–Bruhat space
E860122
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Schwartz–Bruhat space canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10389289 — 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: Schwartz–Bruhat space Context triple: [Weil representation, actsOn, Schwartz–Bruhat space]
-
A.
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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B.
Plancherel theorem for real reductive groups
The Plancherel theorem for real reductive groups is a fundamental result in representation theory that describes how square-integrable functions on a real reductive Lie group decompose into irreducible unitary representations, generalizing Fourier analysis to this non-abelian setting.
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C.
Introduction to Abstract Harmonic Analysis
Introduction to Abstract Harmonic Analysis is a foundational graduate-level textbook that systematically develops the theory of harmonic analysis on topological groups and related abstract structures.
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D.
Bourgain spaces
Bourgain spaces are function spaces introduced by Jean Bourgain that are tailored to study the well-posedness and regularity of nonlinear dispersive partial differential equations.
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E.
Gelfand transform
The Gelfand transform is a fundamental construction in functional analysis that represents elements of a commutative Banach algebra as continuous functions on its space of maximal ideals, linking algebraic structure with topological and spectral properties.
- 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: Schwartz–Bruhat space Target entity description: 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.
-
A.
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.
-
B.
Plancherel theorem for real reductive groups
The Plancherel theorem for real reductive groups is a fundamental result in representation theory that describes how square-integrable functions on a real reductive Lie group decompose into irreducible unitary representations, generalizing Fourier analysis to this non-abelian setting.
-
C.
Introduction to Abstract Harmonic Analysis
Introduction to Abstract Harmonic Analysis is a foundational graduate-level textbook that systematically develops the theory of harmonic analysis on topological groups and related abstract structures.
-
D.
Bourgain spaces
Bourgain spaces are function spaces introduced by Jean Bourgain that are tailored to study the well-posedness and regularity of nonlinear dispersive partial differential equations.
-
E.
Gelfand transform
The Gelfand transform is a fundamental construction in functional analysis that represents elements of a commutative Banach algebra as continuous functions on its space of maximal ideals, linking algebraic structure with topological and spectral properties.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
LF-space
ⓘ
function space ⓘ nuclear space ⓘ test function space ⓘ topological vector space ⓘ |
| characterizedBy |
compact support in the non-Archimedean case
ⓘ
local constancy in the non-Archimedean case ⓘ rapid decay ⓘ smoothness in the Archimedean case ⓘ |
| closedUnder |
Fourier transform
NERFINISHED
ⓘ
convolution ⓘ multiplication by characters ⓘ translations ⓘ |
| definedOn |
LCA group
ⓘ
adele ring ⓘ finite-dimensional complex vector space ⓘ finite-dimensional real vector space ⓘ locally compact abelian group ⓘ non-Archimedean local field ⓘ p-adic field ⓘ real vector space ⓘ |
| dualSpace | space of tempered distributions on the group ⓘ |
| elementType |
complex-valued functions
ⓘ
test functions ⓘ |
| field |
adelic analysis
ⓘ
automorphic forms ⓘ harmonic analysis ⓘ number theory ⓘ representation theory ⓘ |
| generalizes |
Schwartz space
NERFINISHED
ⓘ
space of rapidly decreasing smooth functions on R^n ⓘ |
| hasProperty |
Montel space in many cases
ⓘ
dense in L^1 of the underlying group ⓘ dense in L^2 of the underlying group ⓘ invariant under Fourier transform up to normalization ⓘ nuclear in the sense of Grothendieck ⓘ |
| introducedInContext |
adelic formulation of number theory
ⓘ
harmonic analysis on local fields ⓘ |
| namedAfter |
François Bruhat
NERFINISHED
ⓘ
Laurent Schwartz NERFINISHED ⓘ |
| topology |
inductive limit topology of Fréchet spaces in many cases
ⓘ
locally convex topology ⓘ |
| usedFor |
Poisson summation formula
NERFINISHED
ⓘ
Tate’s thesis NERFINISHED ⓘ Weil representation NERFINISHED ⓘ definition of Fourier transform on LCA groups ⓘ distribution theory on LCA groups ⓘ local zeta integrals ⓘ theory of automorphic L-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: Schwartz–Bruhat space Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.