On lineability of sets of unimaximal continuous functions.

Lucas Quarta

Université de Mons-Hainaut, Belgium.

A set M in a linear topological space X is said to be n-lineable (resp. lineable, resp. spaceable) in X if M ∪ {0} contains a vector space of dimension n (resp. infinite dimensional, resp. closed and infinite dimensional). If the maximum cardinality of such a vector space exists it is called the lineability of M. One can find in [1] a lot of results of “linearity in non linear problems” in many different fields of analysis. One of the first works in this spirit is the lineability of the set of nowhere differentiable functions on [0, 1],[6] (see also [4], [7] and [8]). Recently, several papers were devoted to the study of the lineability of sets of functions on [0, 1] or ℝ which satisfy other special properties (see for example [2] and [3]). Our goal is the study of the lineability of the set M = Ĉ(S) of continuous functions defined on a set S and which admit one and only one absolute maximum. We will present an analytical approach when S is a subset of ℝ (joint work with V.I. Gurariy) and a geometrical one (based on ideas introduced in [5]) in higher dimensions (joint work with C. Troestler).

References

[1] R. Aron, D. Garcia, M. Maestre, Linearity in non linear problems, Rev. R. Acad. Cien. Serie A. Mat. Vol. 95(1), 2001, 7–12.

[2] R. Aron, V. I. Gurariy, J. Seoane, Lineability and spaceability of sets of functions on ℝ, to appear in Proc. Amer. Math. Soc.

[3] P. Enflo, V. I. Gurariy, On lineability and spaceability of sets in function spaces, preprint.

[4] V. Fonf, V. I. Gurariy, V. Kadec, An infinite dimensional subspace of C[0,1] consisting of nowhere differentiable functions, C. R. Acad. Bulgare Sci. 52 (1999), No 11–12, 13–16.

[5] G. Godefroy, J. Saint Raymond, Sous-espaces de dimension 3 des espaces C(K), Bull. Sc. math., 2ème série, 103 (1979), 259–262.

[6] V. I. Gurariy, subspaces and bases in spaces of continuous functions, (Russian) Dokl. Akad. Nauk SSSR 167 (1966), 971–973.

[7] V. I. Gurariy, Linear spaces composed of nondifferentiable functions, C. R. Acad. Bulgare Sci. 44 (1991), No 5, 13–16.

[8] L. Rodriguez-Piazza, Every separable Banach space is isometric to a space of continuous nowhere differentiable functions, Proc. Amer. Math. Soc. 123, 12 (1995), 3649–3654.