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Lineability of non-differentiable Pettis primitives

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Abstract

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 \(\mathbf{ND}\).

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Authors and Affiliations

  1. Department of Mathematics, University of Palermo, Via Archirafi 34, 90123, Palermo, Italy

    B. Bongiorno & L. Di Piazza

  2. Department of Mathematics, University of Louisville, Louisville, KY, 40292, USA

    U. B. Darji

Authors
  1. B. Bongiorno
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  2. U. B. Darji
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  3. L. Di Piazza
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Correspondence to U. B. Darji.

Additional information

Communicated by G. Teschl.

U. B. Darji would like to thank the hospitality of the Department of Mathematics of University of Palermo and grant Cori 2013 of the University of Palermo.

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Bongiorno, B., Darji, U.B. & Di Piazza, L. Lineability of non-differentiable Pettis primitives. Monatsh Math 177, 345–362 (2015). https://doi.org/10.1007/s00605-014-0703-6

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  • DOI: https://doi.org/10.1007/s00605-014-0703-6

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