A pairwise almost disjoint subspace of kN of maximal dimension

We show that if k is an infinite field, then there exists a subspace $W?k^{\mathbb{N}}$ of dimension $|k{|}^{{\mathrm{?}}_0}$, such that no nonzero member of W has infinitely many zeros. This generalizes a result from a paper by Bergman and Nahlus, and partly answers another question from the same paper.