Lebesgue spaces

E87728

Banach space family function space family mathematical concept

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.

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All labels observed (4)

Label	Occurrences
Lebesgue space	2
Lebesgue spaces canonical	2
L^p spaces	1
Lebesgue space L^p	1

Statements (47)

Predicate	Object
instanceOf	Banach space family ⓘ function space family ⓘ mathematical concept ⓘ
alsoKnownAs	Lp spaces ⓘ
application	Fourier analysis ⓘ ergodic theory ⓘ harmonic analysis ⓘ interpolation theory ⓘ partial differential equations ⓘ probability theory ⓘ
basedOn	Lebesgue measure ⓘ measure space ⓘ
definedOn	measure space (X, Σ, μ) ⓘ
definingCondition	f is essentially bounded for p = ∞ ⓘ \|f\|^p is integrable ⓘ ∫ \|f\|^p dμ < ∞ for 1 ≤ p < ∞ ⓘ
duality	(L^p)* ≅ L^q for 1 < p < ∞ and 1/p + 1/q = 1 ⓘ
elementType	equivalence classes of measurable functions ⓘ
equivalenceRelation	equality almost everywhere ⓘ
field	functional analysis ⓘ measure theory ⓘ probability theory ⓘ
inequality	Hölder inequality holds in L^p spaces ⓘ Minkowski inequality holds in L^p spaces ⓘ
introducedBy	Henri Lebesgue ⓘ
L1Definition	integrable functions ⓘ
L1Dual	L^∞ in many standard measure spaces ⓘ
L2Definition	square-integrable functions ⓘ
L2InnerProduct	∫ f·conjugate(g) dμ ⓘ
L2Is	Hilbert space ⓘ
LInfinityDefinition	essentially bounded measurable functions ⓘ
LInfinityDual	larger than L^1 in general ⓘ
LpDefinition	p-integrable functions ⓘ
LpInclusion	L^q ⊆ L^p under suitable measure conditions when q > p ⓘ
norm	(∫ \|f\|^p dμ)^{1/p} for 1 ≤ p < ∞ ⓘ essential supremum norm for p = ∞ ⓘ
notation	L^p ⓘ
parameter	p ⓘ
parameterRange	1 ≤ p ≤ ∞ ⓘ
property	Banach spaces for 1 ≤ p ≤ ∞ ⓘ complete normed spaces ⓘ
relatedConcept	Banach spaces ⓘ surface form: Banach function spaces Orlicz spaces ⓘ Sobolev spaces ⓘ
role	fundamental framework in modern analysis ⓘ standard setting for random variables in probability theory ⓘ
specialCase	Hilbert space when p = 2 ⓘ

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Minkowski inequality → holdsIn → Lebesgue spaces ⓘ

Henri Lebesgue → notableConcept → Lebesgue spaces ⓘ

this entity surface form: Lebesgue space

Young inequality for convolutions → involvesConcept → Lebesgue spaces ⓘ

this entity surface form: Lebesgue space

Sobolev spaces → relatedConcept → Lebesgue spaces ⓘ

this entity surface form: Lebesgue space L^p

Orlicz spaces → generalizes → Lebesgue spaces ⓘ

this entity surface form: L^p spaces