Abstract. It is proved the existence of large algebraic structures -including large vector subspaces or infinitely generated free algebrasinside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of non-continuous separately continuous real functions. Lineability in special spaces of sequences is also investigated. Some of our findings complete or extend a number of results by several authors.

We link the overconvergence properties of certain Taylor series in the unit disk to the maximality of their cluster sets, so connecting outer wild behavior to inner wild behavior. Specifically, it is proved the existence of a dense linear manifold of holomorphic functions in the disk that are, except for zero, universal Taylor series in the sense of Nestoridis and, simultaneously, have maximal cluster sets along many curves tending to the boundary. Moreover, it is constructed a dense linear manifold of universal Taylor series having, for each boundary point, limit zero along some path which is tangent to the corresponding radius. Finally, it is proved the existence of a closed infinite dimensional manifold of holomorphic functions enjoying the two-fold wild behavior specified at the beginning.

The existence of a dense linear manifold of holomorphic functions on a Jordan domain having except for zero maximal cluster set along any curve tending to the boundary with nontotal oscillation value set is shown. r 2004 Elsevier Inc. All rights reserved. MSC: primary 30D40; secondary 30E10; 30H05

Let Ω be a domain in the N -dimensional real space, L be an elliptic differential operator, and (T n ) be a sequence whose members belong to a certain class of operators defined on the space of L-analytic functions on Ω. It is proved in this paper the existence of a dense linear manifold of L-analytic functions all of whose nonzero members have maximal cluster sets under the action of every T n along any curve ending at the boundary of Ω such that its ω-limit does not contain any component of the boundary. The above class contains all partial differentiation operators ∂ α , hence the statement extends earlier results due to Boivin, Gauthier and Paramonov, and to the first, third and fourth authors. Key words and phrases: maximal cluster set, L-analytic function, dense linear manifold, admissible path, elliptic operator, internally controlled operator.

In this paper, a study of topological and algebraic properties of two families of functions from the unit interval I into the plane R 2 is performed. The first family is the collection of all Peano curves, that is, of those continuous mappings onto the unit square. The second one is the bigger set of all spacefilling curves, i.e. of those continuous functions I → R 2 whose images have the positive Jordan content. Emphasis is put on the size of these families, in both topological and algebraic senses, when endowed with natural structures.