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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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 ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Weil representation → actsOn → Schwartz–Bruhat space ⓘ