Lebesgue spaces
E87728
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.
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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.