L^p(X, μ)

GPTKB entity

Predicate	Object
gptkbp:instanceOf	function space
gptkbp:alsoKnownAs	gptkb:Lp_space
gptkbp:complete	yes
gptkbp:consistsOf	measurable functions
gptkbp:definedIn	measure space (X, μ)
gptkbp:elementCondition	∫\|f\|^p dμ < ∞
gptkbp:generalizes	ℓ^p spaces
gptkbp:hasDual	L^q(X, μ) where 1/p + 1/q = 1, for 1 < p < ∞
gptkbp:hasSpecialCase	L^1(X, μ) is the space of integrable functions L^2(X, μ) is a Hilbert space L^∞(X, μ) is the space of essentially bounded functions
gptkbp:hasSubgroup	measurable functions on X
https://www.w3.org/2000/01/rdf-schema#label	L^p(X, μ)
gptkbp:importantFor	gptkb:Hölder's_inequality gptkb:Fatou's_lemma gptkb:Riesz_representation_theorem gptkb:Minkowski's_inequality Lebesgue dominated convergence theorem
gptkbp:introduced	gptkb:Henri_Lebesgue
gptkbp:isBanachSpace	yes
gptkbp:isHilbertSpace	if p = 2
gptkbp:norm	p-norm
gptkbp:normFormula	(∫\|f\|^p dμ)^{1/p}
gptkbp:parameter	p ≥ 1
gptkbp:usedIn	gptkb:probability_theory functional analysis harmonic analysis
gptkbp:vectorSpaceOver	complex numbers real numbers
gptkbp:zeroElement	zero function
gptkbp:bfsParent	gptkb:Lebesgue_space
gptkbp:bfsLayer	6