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A subset A of a vector space X is called \alpha-lineable whenever A contains, except for the null vector, a subspace of dimension \alpha. If X has a topology, then A is \alpha-spaceable if such subspace can be chosen to be closed. The vast existing literature on these topics has shown that positive results for lineability and spaceability are quite common. Recently, the stricter notions of (\alpha,\beta)-lineability/spaceability were introduced as an attempt to shed light on more subtle issues. In this paper, among other results, we prove some general criteria for the notion of (\alpha,\beta)-lineability/spaceability and, as applications, we extend recent results of different authors. [ABSTRACT FROM AUTHOR]

For each vector x\in \ell ^{\infty }, we can define the non-empty compact set L_x of accumulation points of x. Given an infinite subset A of \mathbb {N}\backslash \{1\}, we can therefore investigate under which conditions on A, the set L(A)≔\{x\in \ell ^\infty : |L_x|\in A\} is lineable or even densely lineable. In particular, we show that if L(A) is lineable then there exists k\ge 1 such that A\cap (A-k) is infinite and that if L(A) is densely lineable then A\cap (A-1) is infinite. We end up by answering an open question on the existence of a closed non-separable subspace in which each non-zero vector has countably many accumulation points. [ABSTRACT FROM AUTHOR]

K. Weierstrass (1872) was probably the first to present the existence of continuous nowhere differentiable functions (although B. Bolzano, in 1822, was the first to come up with such a construction). Almost a century later, V. Gurariĭ [Dokl. Akad. SSSR 167 (1966), pp. 971–973] observed that the family of continuous functions on [0,1] that are differentiable at no point contains, except for the null function, an infinite dimensional vector space. Moreover, and among other recent contributions in this direction, S. Hencl [Proc. Amer. Math. Soc. 128 (2000), p. 3505–3511] generalized the previously mentioned result by proving the existence of isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions. Here, we continue this ongoing research with the study of continuous nowhere Hölder functions, no longer defined in subsets of \mathbb {R}, but in subsets of the p-adic field \mathbb {Q}_p. [ABSTRACT FROM AUTHOR]

We show that the set \ell ^{\infty }\setminus c_0 is maximal dense-lineable and densely strongly \mathfrak {c}-algebrable answering a question posed by Nestoridis and complementing a result by García-Pacheco, Martín and Seoane-Sepúlveda. [ABSTRACT FROM AUTHOR]

We introduce the class of everywhere like functions, which helps us to recover some known classes (such as that of everywhere surjective ones). We also study the algebraic genericity of this new class together with the class of fractal functions. [ABSTRACT FROM AUTHOR]

We investigate the additivity A and lineability L cardinal coefficients for the following classes of functions: ES \ SES of everywhere surjective functions that are not strongly everywhere surjective, Darboux-like, Sierpiński-Zygmund, surjective, and their corresponding intersections. The classes SES and ES have been shown to be 2c-lineable. In contrast, although we prove here that ES \ SES is c+-lineable, it is still unclear whether it can be proved in ZFC that ES \ SES is 2c-lineable. Moreover, we prove that if c is a regular cardinal number, then A(ES \ SES) ≤ c. This shows that, for the class ES \ SES, there is an unusually large gap between the numbers A and L. [ABSTRACT FROM AUTHOR]