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In this paper, we show that the set of continuous functions defined on R n that approach zero at infinity and attain their maximum at precisely one (and only one) point is n-lineable but not (n + 2) -lineable. This result complements some recent published works on an open question originally posed by Vladimir I. Gurariy (1935–2005) in 2003. [ABSTRACT FROM AUTHOR]

Important probabilistic problems require to find the limit of a sequence of random variables. However, this limit can be understood in different ways and various kinds of convergence can be defined. Among the many types of convergence of sequences of random variables, we can highlight, for example, that convergence in L p -sense implies convergence in probability, which, in turn, implies convergence in distribution, besides that all these implications are strict. In this paper, the relationship between several types of convergence of sequences of random variables will be analyzed from the perspective of lineability theory. [ABSTRACT FROM AUTHOR]

In this paper we introduce the concept of infinite pointwise dense lineability (spaceability), and provide a criterion to obtain density from mere lineability. As an application, we study the linear and topological structures within the set of infinite differentiable and integrable functions, for any order p ≥ 1 , on R N which are unbounded in a pre-fixed set. [ABSTRACT FROM AUTHOR]

Since the concept of lineability was coined by Gurariy in the early 2000s, only a few results provide a general criterion that guarantees the existence of large algebraic structures. Furthermore, most results in lineability theory are generally positive results, and there are a small amount of negative results. In this paper, we provide a non-constructive general (α , β) -lineability criterion in the context of complements of vector spaces, and we present two criteria of non- (α , β) -spaceability. Several applications are presented. [ABSTRACT FROM AUTHOR]

Recently Tomasz Natkaniec in (Natkaniec in Rev R Acad Cenc Exactas Fís Nat Ser A Mat RACSAM, 115(1):10, 2021) studied the lineability problem for several classes of non-measurable functions in two variables. In this note we improve his results in the direction of algebrability. In particular, we show that most of the classes considered by Natkaniec contain free algebras with 2 c many generators. [ABSTRACT FROM AUTHOR]