Infinite Lineability: On the Abundance of Dense Subspaces

In this paper, we investigate the concept of infinite dense-lineability recently introduced by M. Calderón-Moreno, P. Gerlach-Mena and J. Prado-Bassas. We answer a question posed by the authors about the equivalence between infinite (pointwise) dense-lineability and (pointwise) dense-lineability. We prove that the equivalence always holds in first-countable topological vector spaces and under some assumptions about the weight of the topology. However, the equivalence is not always true, as shown in an example. Furthermore, we introduce the notions of infinite $(\alpha ,\beta )$-dense-lineability and infinite (strongly) dense-algebrability and obtain some analogous results in these cases. We also obtain a criterion for strongly dense-algebrability for sets of the form $X\setminus Y$, where $X$ is a free algebra and $Y$ is a free subalgebra of $X$.