Journal of Convex Analysis 30 (2023), No. 1, 065--080
Copyright Heldermann Verlag 2023
On Special Partitions of [0,1] and Lineability within Families of Bounded Variation Functions
Luis Bernal-González
Dep. de Análisis Matemático, Facultad de Matemáticas, Inst. de Matemáticas Antonio de Castro Brzezicki, Universidad de Sevilla, Spain
lbernal@us.es
Juan Fernández-Sánchez
Universidad de Almería, Spain
juanfernandez@ual.es
Juan B. Seoane-Sepulveda
Inst. de Matemática Interdisciplinar, Dep. de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Spain
jseoane@ucm.es
Wolfgang Trutschnig
Dept. of Mathematics, University of Salzburg, Austria
wolfgang@trutschnig.net
We show that there exist large algebraic structures (vector spaces, algebras, closed subspaces, etc.) formed entirely (except for 0), on one hand, by singular, nowhere monotonic functions on [0,1] and, on the other hand, by absolutely continuous nowhere monotonic functions. Several tools, of independent interest, related to obtaining special partitions of R into uncountable collections will be provided and used. The results obtained in this note are either new or improved versions of already existing ones.
Keywords: Lineability, spaceability, bounded variation function, singular function, absolutely continuous function, nowhere monotonic function.
MSC: 15A03, 46B87, 26A45, 26B30.
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