Statements (47)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:alsoKnownAs |
gptkb:Lebesgue_spaces
|
| gptkbp:application |
gptkb:probability_theory
gptkb:signal_processing harmonic analysis partial differential equations |
| gptkbp:basisFor |
do not generally have a Schauder basis
|
| gptkbp:definedIn |
1 ≤ p ≤ ∞
|
| gptkbp:defines |
space of measurable functions whose p-th power is Lebesgue integrable
|
| gptkbp:field |
gptkb:mathematics
functional analysis |
| gptkbp:generalizes |
sequence spaces l^p
|
| gptkbp:hasSpecialCase |
L^∞ is the space of essentially bounded functions
L^2 is a Hilbert space L^1 is the space of absolutely integrable functions |
| https://www.w3.org/2000/01/rdf-schema#label |
Lp spaces
|
| gptkbp:introduced |
gptkb:Henri_Lebesgue
|
| gptkbp:namedAfter |
gptkb:Henri_Lebesgue
|
| gptkbp:norm |
||f||_p = (∫ |f|^p)^{1/p} for 1 ≤ p < ∞
||f||_∞ = ess sup |f| |
| gptkbp:notation |
L^p
|
| gptkbp:property |
complete normed vector space
Banach space for 1 ≤ p ≤ ∞ L^1 dual is L^∞ L^2 has an inner product L^p is a measure space dependent L^p is a metric space L^p is a normed space L^p is a topological vector space L^p is dilation invariant L^p is not a Hilbert space for p ≠ 2 L^p is translation invariant L^p is used in ergodic theory L^p is used in interpolation theory L^p is used in machine learning L^p is used in quantum mechanics L^p is used in statistics L^∞ dual is not L^1 L^p is a vector space over the complex or real numbers reflexive for 1 < p < ∞ separable for 1 ≤ p < ∞ dual of L^p is L^q where 1/p + 1/q = 1, 1 < p < ∞ |
| gptkbp:relatedTo |
gptkb:Sobolev_spaces
Fourier analysis |
| gptkbp:bfsParent |
gptkb:Hölder's_inequality
gptkb:Minkowski's_inequality |
| gptkbp:bfsLayer |
5
|