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Searching for structures inside of the family of bounded derivatives which are not Riemann integrable. (English) Zbl 1450.46014

Summary: We construct a non-separable Banach space every nonzero element of which is a bounded derivative that is not Riemann integrable. This in particular improves a result presented in [D. García et al., Math. Nachr. 283, No. 5, 712–720 (2010; Zbl 1210.46019)], where the corresponding space was found to be separable.

MSC:

46B87 Lineability in functional analysis
46E15 Banach spaces of continuous, differentiable or analytic functions
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A36 Antidifferentiation
15A03 Vector spaces, linear dependence, rank, lineability

Citations:

Zbl 1210.46019
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Full Text: DOI Euclid

References:

[1] R. M. Aron, L. Bernal-González, D. M. Pellegrino and J. B. Seoane-Sepúlveda, Lineability: The Search for Linearity in Mathematics, Monographs and Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2016. · Zbl 1348.46001
[2] R. M. Aron, V. I. Gurariy and J. B. Seoane-Sepúlveda, Lineability and spaceability of sets of functions on \(\mathbb{R} \), Proc. Amer. Math. Soc. 133 (2005), no. 3, 795-803. · Zbl 1069.26006
[3] D. García, B. C. Grecu, M. Maestre and J. B. Seoane-Sepúlveda, Infinite dimensional Banach spaces of functions with nonlinear properties, Math. Nachr. 283 (2010), no. 5, 712-720. · Zbl 1210.46019
[4] R. A. Gordon, A bounded derivative that is not Riemann integrable, Math. Mag. 89 (2016), no. 5, 364-370. · Zbl 1398.26002 · doi:10.4169/math.mag.89.5.364
[5] E. M. Granath, Bounded Derivatives Which are not Riemann Integrable, Thesis (Masters)–Whitman College, 2017.
[6] P. Jiménez-Rodríguez, \(c_0\) is isometrically isomorphic to a subspace of Cantor-Lebesgue functions, J. Math. Anal. Appl. 407 (2013), no. 2, 567-570. · Zbl 1319.46022
[7] O. A. Nielsen, An Introduction to Integration and Measure Theory, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, 1997. · Zbl 0882.28001
[8] J. B. Seoane-Sepúlveda, Chaos and Lineability of Pathological Phenomena in Analysis, Thesis (Ph.D.)–Kent State University, 2006.
[9] V. Volterra, Sui principii del calcolo integrale, Giorn. Mat. Battaglini 19 (1881), 333-372. · JFM 13.0213.02
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