Bochner integral

E613404

generalization of Lebesgue integral integral mathematical concept

The Bochner integral is a generalization of the Lebesgue integral to functions taking values in Banach spaces, widely used in functional analysis and probability theory.

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Observed surface forms (1)

Surface form	Occurrences
Radon integral	1

Statements (48)

Predicate	Object
instanceOf	generalization of Lebesgue integral ⓘ integral ⓘ mathematical concept ⓘ
appliesTo	Banach space–valued functions ⓘ
assumes	complete measure space for many theorems ⓘ
category	vector-valued integration theory ⓘ
closedUnder	almost everywhere limits under dominated convergence ⓘ finite linear combinations of integrable functions ⓘ
codomain	Banach space ⓘ
compatibleWith	classical Lebesgue integral on scalar functions ⓘ
contrastWith	Pettis integral in nonreflexive spaces ⓘ Riemann integral NERFINISHED ⓘ
definitionUses	norm in Banach space ⓘ simple functions ⓘ
domain	measure space ⓘ
extends	Lebesgue integral for complex-valued functions ⓘ Lebesgue integral for real-valued functions ⓘ
field	functional analysis ⓘ measure theory ⓘ probability theory ⓘ
generalizes	Lebesgue integral NERFINISHED ⓘ
implies	Pettis integrability ⓘ
integrandCondition	norm is Lebesgue integrable ⓘ
introducedBy	Salomon Bochner NERFINISHED ⓘ
linearity	is linear in the integrand ⓘ
namedAfter	Salomon Bochner NERFINISHED ⓘ
notation	∫ f dμ for Banach-valued f ⓘ
relatedTo	Dunford integral ⓘ Pettis integral NERFINISHED ⓘ
requires	Bochner measurability ⓘ almost separably valued functions for strong measurability ⓘ integrability of norm ⓘ separability of essential range for strong measurability ⓘ strong measurability ⓘ
satisfies	Fubini theorem for Banach-valued functions NERFINISHED ⓘ dominated convergence theorem (Bochner version) NERFINISHED ⓘ monotone convergence theorem for nonnegative scalar norms ⓘ
strongerThan	Pettis integrability ⓘ
targetSpace	Banach space ⓘ complete normed vector space ⓘ
timeOfIntroduction	20th century ⓘ
usedIn	evolution equations ⓘ operator-valued integration ⓘ partial differential equations with Banach-valued data ⓘ semigroup theory ⓘ stochastic integration in Banach spaces ⓘ vector-valued Lp spaces ⓘ
yields	Bochner Lp spaces NERFINISHED ⓘ

Referenced by (2)

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

Salomon Bochner → notableFor → Bochner integral ⓘ

Johann Radon → knownFor → Bochner integral ⓘ

this entity surface form: Radon integral