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We show that, when omitting one condition in several well‐known convergence results from probability and measure theory (such as the Dominated Convergence Theorem, Fatou's Lemma, or the Strong Law of Large Numbers), we can construct "very large" (in terms of the cardinality of their systems of generators) spaces and algebras of counterexamples. Moreover, we show that on the probability space ([0,1],B([0,1]),λ)$([0,1],\mathcal {B}([0,1]),\lambda)$ the families of sequences of random variables converging in probability but (i) not converging outside a set of measure 0 or (ii) not converging in arithmetic mean are also "very large". [ABSTRACT FROM AUTHOR]

Giving a partial answer to a conjecture formulated by Aron, Boyd, Ryan and Zalduendo, we show that if a real Banach space X is not linearly and continuously injected into a Hilbert space, then for any 2-homogeneous continuous polynomial P on X, its zero-set is not separable. For this class of spaces, we also prove that, if P is semidefinite, then is a nonseparable linear subspace. [ABSTRACT FROM AUTHOR]

We employ a classical result by Toeplitz (1913) and the seminal work by Bohnenblust and Hille on Dirichlet series (1931) to show that the set of continuous m-homogeneous non-analytic polynomials on c0 contains an isomorphic copy of ℓ1. Moreover, we can have this copy of ℓ1 in such a way that every non-zero element of it fails to be analytic at precisely the same point. [ABSTRACT FROM AUTHOR]

Abstract: Let X be a sequence space and denote by the subset of X formed by sequences having only a finite number of zero coordinates. We study algebraic properties of and show (among other results) that (for ) does not contain infinite dimensional closed subspaces. This solves an open question originally posed by R.M. Aron and V.I. Gurariy in 2003 on the linear structure of . In addition to this, we also give a thorough analysis of the existing algebraic structures within the sets and and their algebraic genericities. [Copyright &y& Elsevier]

A subset M of a topological vector space X is called lineable (respectively, spaceable) in X if there exists an infinite dimensional linear space (respectively, an infinite dimensional closed linear space) Y ⊂ M ∪ {0}. In this article we prove that, for every infinite dimensional closed subspace X of C[0, 1], the set of functions in X having infinitely many zeros in [0, 1] is spaceable in X. We discuss problems related to these concepts for certain subsets of some important classes of Banach spaces (such as C[0, 1] or Müntz spaces). We also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of C[0, 1], as well as oscillating and annulling properties of subspaces of C[0, 1]. [ABSTRACT FROM AUTHOR]

For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed sub-spaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability. [ABSTRACT FROM AUTHOR]

Abstract: Recently, it has been proved that, assuming that there is an almost disjoint family of cardinality in (which is assured, for instance, by either Martin’s Axiom, or the Continuum Hypothesis, or even ) one has that the set of Sierpiński–Zygmund functions is -strongly algebrable (and, thus, -lineable). Here we prove that these two statements are actually equivalent and, moreover, that they both are undecidable. This would be the first time in which one encounters an undecidable proposition in the recently coined theory of lineability and spaceability. [Copyright &y& Elsevier]

In this note, the linear structure of the family H e ( G ) of holomorphic functions in a domain G of a complex Banach space that are not holomorphically continuable beyond the boundary of G is analyzed. More particularly, we prove that H e ( G ) contains, except for zero, a closed (and a dense) vector space having maximal dimension, as well as a maximally generated free algebra. The results obtained complete a number of previous ones by several authors. [ABSTRACT FROM AUTHOR]

Abstract: Let be a non-compact locally compact group. In this paper we study the size of the set where and are normed spaces of continuous functions on . We also consider the problem of the spaceability of the set and (among other results) we show that, for , the above set is strongly -algebrable (and, therefore, algebrable and lineable) with respect to the convolution product. [Copyright &y& Elsevier]

Abstract: We give a condition for a family of functions to be lineable by means of its additivity. The novelty presented here is that we solve the lineability problem using a technique that is not constructive, as are most approaches to this problem. We relate the notions of additivity and lineability and use this relation to give a general method to find the lineability of large families of functions. We also study more examples of pathologically behaving functions, in particular, the class of Jones functions, which is a highly pathological subclass of perfectly everywhere surjective functions. We work on the additivity, lineability, and main properties of this class. [Copyright &y& Elsevier]

Abstract: We introduce the concept of maximal lineability cardinal number, , of a subset M of a topological vector space and study its relation to the cardinal numbers known as: additivity , homogeneous lineability , and lineability of M. In particular, we will describe, in terms of , the lineability and spaceability of the families of the following Darboux-like functions on , : extendable, Jones, and almost continuous functions. [Copyright &y& Elsevier]

Abstract: In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the lineability of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered. [Copyright &y& Elsevier]

We show that there exists an uncountably generated algebra every nonzero element of which is an everywhere surjective function on C, that is, a function f : C → C such that, for every non void open set U ⊂ C, f (U) = C. [ABSTRACT FROM AUTHOR]