Schwartz–Bruhat space

E860122

LF-space function space nuclear space test function space topological vector 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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All labels observed (2)

Label	Occurrences
Fourier transform on local fields	1
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.
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
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 ⓘ 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 ⓘ 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 ⓘ Laurent Schwartz ⓘ
topology	inductive limit topology of Fréchet spaces in many cases ⓘ locally convex topology ⓘ
usedFor	Poisson summation formula ⓘ Tate’s thesis ⓘ linked to: Tate's thesis Weil representation ⓘ definition of Fourier transform on LCA groups ⓘ distribution theory on LCA groups ⓘ local zeta integrals ⓘ theory of automorphic L-functions ⓘ