6533b862fe1ef96bd12c6436

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Lineability of non-differentiable Pettis primitives

Udayan B. DarjiB. BongiornoL. Di Piazza

subject

Discrete mathematicsPettis integralMathematics::Functional AnalysisIntegrable systemGeneral MathematicsBanach space46G10 28B05Functional Analysis (math.FA)Mathematics - Functional AnalysisSet (abstract data type)Dvoretzky's theoremFOS: MathematicsLocally integrable functionDifferentiable functionPettis Integral nowhere differentiable Dvoretzky's theorem lineable spaceableMathematicsVector space

description

Let \(X\) be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an \(X\)-valued Pettis integrable function on \([0,1]\) whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that \(\mathbf{ND}\), the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to \(\mathbf{ND}\).

yearjournalcountryeditionlanguage
2014-03-07Monatshefte für Mathematik
https://doi.org/10.1007/s00605-014-0703-6