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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Schwartz space S(ℝⁿ) ⊂ L²(ℝⁿ) ⊂ S′(ℝⁿ) | 1 |
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 ⓘ |
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)