Moduleability, algebraic structures, and nonlinear properties
This paper deals with the notions of lineability and spaceability of sets of functions, i.e., with the search of large linear structures inside sets of pathological functions. A subset \(M\) of functions is lineable if \(M\cup \{0\}\) contains an infinite-dimensional vector space and it is spaceable if it contains a closed infinite-dimensional vector space. Analogously, the set is algebrable if it contains an infinitely generated algebra, coneable if it possesses a cone containing an infinite linearly independent set, moduleable. We refer to [\textit{R. Aron, D. García} and \textit{M. Maestre}, RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 95, No.~1, 7--12 (2001; Zbl 1036.46033); \textit{R. Aron, V. I. Gurariy} and \textit{J. B. Seoane}, Proc. Am. Math. Soc. 133, No.~3, 795--803 (2005; Zbl 1069.26006); \textit{V. I. Gurariy} and \textit{L. Quarta}, J. Math. Anal. Appl. 294, No.~1, 62--72 (2004; Zbl 1053.46014)] as the seminal papers of the theory of lineability. This topic has grown a lot in the last years and there are many interesting results, some of which are due to the authors of the paper under review. In the second section of the paper, the authors construct infinite-dimensional vector spaces of non-integrable, measurable functions. Section 3 is dedicated to construct dense and not barrelled subspaces of Banach spaces with Schauder basis every nonzero element of which has a finite number of zero coordinates. Finally, in the last section, pathologies in sets of differentiable functions are investigated.
scientific article; zbMATH DE number 5757945
scientific article