Taiwanese Journal of Mathematics

Searching for Structures Inside of the Family of Bounded Derivatives Which are not Riemann Integrable

Pablo Jiménez-Rodríguez

Full-text: Open access

Abstract

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 [3], where the corresponding space was found to be separable.

Article information

Source
Taiwanese J. Math., Volume 22, Number 6 (2018), 1427-1433.

Dates
Received: 3 December 2017
Accepted: 13 April 2018
First available in Project Euclid: 18 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1524038539

Digital Object Identifier
doi:10.11650/tjm/180403

Mathematical Reviews number (MathSciNet)
MR3878575

Zentralblatt MATH identifier
07021697

Subjects
Primary: 15A03: Vector spaces, linear dependence, rank 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX] 47L05: Linear spaces of operators [See also 46A32 and 46B28]

Keywords
lineability spaceability derivatives Riemann integrability

Citation

Jiménez-Rodríguez, Pablo. Searching for Structures Inside of the Family of Bounded Derivatives Which are not Riemann Integrable. Taiwanese J. Math. 22 (2018), no. 6, 1427--1433. doi:10.11650/tjm/180403. https://projecteuclid.org/euclid.twjm/1524038539


Export citation

References

  • 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.
  • 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.
  • 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.
  • R. A. Gordon, A bounded derivative that is not Riemann integrable, Math. Mag. 89 (2016), no. 5, 364–370.
  • E. M. Granath, Bounded Derivatives Which are not Riemann Integrable, Thesis (Masters)–Whitman College, 2017.
  • 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.
  • 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.
  • J. B. Seoane-Sepúlveda, Chaos and Lineability of Pathological Phenomena in Analysis, Thesis (Ph.D.)–Kent State University, 2006.
  • V. Volterra, Sui principii del calcolo integrale, Giorn. Mat. Battaglini 19 (1881), 333–372.