Schwartz space
E884919
Fréchet space
Montel space
function space
locally convex space
nuclear space
topological vector space
Schwartz space is the function space of rapidly decreasing smooth functions on Euclidean space, fundamental in distribution theory and Fourier analysis.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Schwartz space canonical | 1 |
| Schwartz space S(ℝⁿ) ⊂ L²(ℝⁿ) ⊂ S′(ℝⁿ) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10772850 — 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 space Context triple: [Produits tensoriels topologiques et espaces nucléaires, relatedConcept, Schwartz space]
-
A.
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.
-
B.
Hardy space
A Hardy space is a function space in complex analysis consisting of holomorphic functions on a domain whose mean values on boundary circles (or lines) are uniformly bounded, playing a central role in harmonic and operator theory.
-
C.
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.
-
D.
Hermite functions
Hermite functions are a family of orthogonal functions built from Hermite polynomials and a Gaussian weight, widely used in quantum mechanics, signal processing, and approximation theory.
-
E.
Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
- 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 space Target entity description: Schwartz space is the function space of rapidly decreasing smooth functions on Euclidean space, fundamental in distribution theory and Fourier analysis.
-
A.
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.
-
B.
Hardy space
A Hardy space is a function space in complex analysis consisting of holomorphic functions on a domain whose mean values on boundary circles (or lines) are uniformly bounded, playing a central role in harmonic and operator theory.
-
C.
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.
-
D.
Hermite functions
Hermite functions are a family of orthogonal functions built from Hermite polynomials and a Gaussian weight, widely used in quantum mechanics, signal processing, and approximation theory.
-
E.
Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Fréchet space
ⓘ
Montel space ⓘ function space ⓘ locally convex space ⓘ nuclear space ⓘ topological vector space ⓘ |
| alsoKnownAs | space of rapidly decreasing smooth functions ⓘ |
| closedUnder |
Fourier transform
NERFINISHED
ⓘ
addition ⓘ differentiation ⓘ multiplication by polynomials ⓘ scalar multiplication ⓘ |
| contains |
Gaussian functions
ⓘ
compactly supported smooth functions ⓘ |
| definedOn |
Euclidean space
ⓘ
ℝ^n ⓘ |
| dualSpace | tempered distributions ⓘ |
| elementCondition |
all derivatives decrease faster than any polynomial grows
ⓘ
infinitely differentiable functions ⓘ rapidly decreasing functions ⓘ |
| field |
Fourier analysis
ⓘ
distribution theory ⓘ functional analysis ⓘ mathematical analysis ⓘ |
| FourierTransform |
is a topological isomorphism on Schwartz space
ⓘ
is an automorphism of Schwartz space ⓘ |
| generalizationOf | space of test functions with compact support in distribution theory ⓘ |
| hasBasisType | countable family of seminorms defining a Fréchet topology ⓘ |
| introducedBy | Laurent Schwartz NERFINISHED ⓘ |
| introducedFor | rigorous theory of distributions ⓘ |
| isDenseIn |
C_0(ℝ^n)
ⓘ
L^p(ℝ^n) for 1 ≤ p < ∞ ⓘ |
| namedAfter | Laurent Schwartz NERFINISHED ⓘ |
| property |
complete
ⓘ
metrizable ⓘ nuclear ⓘ reflexive ⓘ separable ⓘ |
| roleIn |
Fourier transform of tempered distributions
ⓘ
definition of tempered distributions ⓘ |
| subsetOf |
C^∞(ℝ^n)
ⓘ
L^p(ℝ^n) for all 1 ≤ p ≤ ∞ ⓘ |
| symbol |
S(R^n)
ⓘ
S(ℝ^n) ⓘ |
| topologyDefinedBy | countable family of seminorms ⓘ |
| usedIn |
partial differential equations
ⓘ
quantum field theory ⓘ signal processing ⓘ |
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 space Description of subject: Schwartz space is the function space of rapidly decreasing smooth functions on Euclidean space, fundamental in distribution theory and Fourier analysis.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Gelfand triple
this entity surface form:
Schwartz space S(ℝⁿ) ⊂ L²(ℝⁿ) ⊂ S′(ℝⁿ)