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gptkbp:instanceOf
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function space
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gptkbp:alsoKnownAs
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gptkb:Lebesgue_spaces
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gptkbp:application
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gptkb:probability_theory
functional analysis
harmonic analysis
partial differential equations
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gptkbp:complete
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true
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gptkbp:contains
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measurable functions
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gptkbp:definedIn
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measure space
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gptkbp:generalizes
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sequence spaces ℓ^p
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gptkbp:hasDual
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L^q where 1/p + 1/q = 1, 1 < p < ∞
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gptkbp:hasSpecialCase
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L^∞ is the space of essentially bounded functions
L^2 is a Hilbert space
L^1 is the space of integrable functions
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gptkbp:hasSubgroup
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measurable functions
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https://www.w3.org/2000/01/rdf-schema#label
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L^p spaces (1 ≤ p ≤ ∞)
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gptkbp:introduced
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gptkb:Henri_Lebesgue
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gptkbp:isBanachSpace
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true
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gptkbp:isHilbertSpace
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true for p=2
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gptkbp:normFormula
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(∫|f|^p dμ)^{1/p} for 1 ≤ p < ∞
ess sup |f| for p=∞
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gptkbp:normType
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p-norm
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gptkbp:parameter
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p
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gptkbp:property
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gptkb:Hölder's_inequality
gptkb:Minkowski_inequality
gptkb:Fatou's_lemma
gptkb:Riesz–Fischer_theorem
reflexive for 1 < p < ∞
separable for 1 ≤ p < ∞
not separable for p=∞
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gptkbp:range
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1 ≤ p ≤ ∞
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gptkbp:bfsParent
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gptkb:Banach_lattice
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gptkbp:bfsLayer
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7
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