6533b873fe1ef96bd12d54f1
RESEARCH PRODUCT
APPROXIMATION OF BANACH SPACE VALUED NON-ABSOLUTELY INTEGRABLE FUNCTIONS BY STEP FUNCTIONS
L. Di PiazzaK. MusiałB. Bongiornosubject
Sobolev spacePure mathematicsRelatively compact subspaceIntegrable systemGeneral MathematicsNorm (mathematics)Step functionMathematical analysisBounded variationBanach spaceLocally integrable functionMathematicsdescription
AbstractThe approximation of Banach space valued non-absolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock–Kurzweil–Pettis and a Denjoy–Khintchine–Pettis integrable function can be only scalarly approximated in the Alexiewicz norm by a sequence of step functions. In case of Henstock–Kurzweil–Pettis and Denjoy–Khintchine–Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2008-09-01 | Glasgow Mathematical Journal |