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.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Lebesgue spaces canonical | 5 |
| Lebesgue space | 2 |
| L^p spaces | 1 |
| Lebesgue space L^p | 1 |
| Lp spaces | 1 |
| Lᵖ spaces | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T736587 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Lebesgue spaces Context triple: [Minkowski inequality, holdsIn, Lebesgue spaces]
-
A.
Lebesgue integration
Lebesgue integration is a foundational measure-theoretic framework for defining and analyzing integrals, particularly suited to handling limits, convergence, and more general functions than those allowed by Riemann integration.
-
B.
Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
-
C.
Minkowski functional
The Minkowski functional is a mathematical tool in functional analysis that assigns a nonnegative real number to each vector in a vector space based on its position relative to a given convex, balanced, absorbing set, generalizing the notion of a norm.
-
D.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
-
E.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Lebesgue spaces Target entity description: 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.
-
A.
Lebesgue integration
Lebesgue integration is a foundational measure-theoretic framework for defining and analyzing integrals, particularly suited to handling limits, convergence, and more general functions than those allowed by Riemann integration.
-
B.
Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
-
C.
Minkowski functional
The Minkowski functional is a mathematical tool in functional analysis that assigns a nonnegative real number to each vector in a vector space based on its position relative to a given convex, balanced, absorbing set, generalizing the notion of a norm.
-
D.
Cameron–Martin theorem
The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).
-
E.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
- F. None of above. chosen
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 function spaces
ⓘ
linked to:
Banach 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Lebesgue spaces Description of subject: 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.
Referenced by (11)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Lebesgue spaces
linked to: Lebesgue spaces
linked to: Lebesgue spaces
linked to: Lebesgue spaces
linked to: Lebesgue spaces
linked to: Lebesgue spaces