Lineability, differentiable functions and special derivatives
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00103-9 ORIGINAL PAPER
Lineability, differentiable functions and special derivatives J. Fernández‑Sánchez1 · D. L. Rodríguez‑Vidanes2 · J. B. Seoane‑Sepúlveda3 · W. Trutschnig4 Received: 14 July 2020 / Accepted: 13 October 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract The present work either extends or improves several results on lineability of differentiable functions and derivatives enjoying certain special properties. Among many other results, we show that there exist large algebraic structures inside the following sets of special functions: (1) The class of differentiable functions with discontinuous derivative on a set of positive measure, (2) the family of differentiable functions with a bounded, non-Riemann integrable derivative, (3) the family of functions from (0, 1) to ℝ that are not derivatives, or (4) the family of mappings that do not satisfy Rolle’s theorem on real infinite dimensional Banach spaces. Several examples and graphics illustrate the obtained results. Keywords Lineability · Algebrability · Rolle’s theorem · Differentiable function · Derivative
Communicated by M. S. Moslehian. * J. B. Seoane‑Sepúlveda [email protected] J. Fernández‑Sánchez [email protected] D. L. Rodríguez‑Vidanes [email protected] W. Trutschnig [email protected] 1
Grupo de investigación “Teoría de cópulas y aplicaciones”, Universidad de Almería, Carretera de Sacramento s/n, 04120 Almería, Spain
2
Departamento de Análisis y Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
3
Departamento de Análisis y Matemática Aplicada, Facultad de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
4
Department for Mathematics, University Salzburg, Hellbrunnerstrasse 34, 5020 Salzburg, Austria
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J. Fernández‑Sánchez et al.
Mathematics Subject Classification 15A03 · 46B87 · 26A15 · 26A27 · 46J10
1 Introduction and preliminaries For the last two decades, many mathematicians have shown interest in the search for large algebraic structures within nonlinear sets of a topological vector space. This area of research receives the name of lineability, a terminology dating back to the early 2000s, coined by Gurariy (1935–2005) and first introduced in [3, 33]. There has been plenty of work in this direction since its appearance. As a matter of fact, this notion was (just recently) introduced by the American Mathematical Society under the MSC2020 15A03 and 46B87 classification references. Research on lineability has shown to be extremely fruitful, a comprehensive description of these concepts (as well as numerous examples and some general techniques) can be found in, e.g., [1, 2, 7–16, 18, 25, 32]. Definition 1.1 Assume that X is a vector space, that 𝛼 is a cardinal number and that A ⊂ X . Then, A is said to be: • lineable if there is an
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