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gptkbp:instanceOf
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function space
|
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gptkbp:alsoKnownAs
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gptkb:Lebesgue_space
|
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gptkbp:application
|
gptkb:probability_theory
functional analysis
harmonic analysis
partial differential equations
|
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gptkbp:basisFor
|
does not have a basis in general
|
|
gptkbp:category
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gptkb:mathematics
analysis
functional analysis
measure theory
topological vector spaces
|
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gptkbp:complete
|
yes
|
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gptkbp:containsElement
|
measurable functions
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gptkbp:definedIn
|
measure space
|
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gptkbp:dimensions
|
infinite (for nontrivial measure spaces)
|
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gptkbp:firstDefined
|
early 20th century
|
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gptkbp:generalizes
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gptkb:Euclidean_space
sequence spaces ℓ^p
|
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gptkbp:hasDual
|
L^q space (where 1/p + 1/q = 1, 1 < p < ∞)
|
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gptkbp:hasSpecialCase
|
gptkb:L^∞_space
gptkb:L^1_space
gptkb:L^2_space
|
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gptkbp:hasSubgroup
|
measurable functions
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
L^p space
|
|
gptkbp:isBanachSpace
|
yes
|
|
gptkbp:isHilbertSpace
|
if p=2
|
|
gptkbp:measures
|
μ
|
|
gptkbp:namedAfter
|
gptkb:Henri_Lebesgue
|
|
gptkbp:norm
|
Lp norm
|
|
gptkbp:normFormula
|
(∫|f|^p dμ)^{1/p}
|
|
gptkbp:notSeparable
|
gptkb:L^∞_space
|
|
gptkbp:par
|
real number greater than or equal to 1
|
|
gptkbp:parameter
|
p
|
|
gptkbp:relatedConcept
|
gptkb:Lp_ball
gptkb:Orlicz_space
gptkb:Sobolev_space
Lp norm
|
|
gptkbp:separates
|
yes, if 1 ≤ p < ∞
|
|
gptkbp:sigmaAlgebra
|
σ-algebra
|
|
gptkbp:topology
|
norm topology
|
|
gptkbp:usedIn
|
gptkb:machine_learning
gptkb:signal_processing
Fourier analysis
statistics
|
|
gptkbp:vectorSpaceOver
|
complex numbers
real numbers
|
|
gptkbp:zeroFunction
|
included
|
|
gptkbp:bfsParent
|
gptkb:Hilbert_space
|
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gptkbp:bfsLayer
|
4
|