  Recently, a lower bound was established on the size of linear sets in
projective spaces, that intersect a hyperplane in a canonical subgeometry.
There are several constructions showing that this bound is tight. In this
paper, we generalize this bound to linear sets meeting some subspace $\pi$ in a
canonical subgeometry. We obtain a tight lower bound on the size of any
$\mathbb F_q$-linear set spanning $\text{PG}(d,q^n)$ in case that $n \leq q$
and $n$ is prime. We also give constructions of linear sets attaining equality
in the former bound, both in the case that $\pi$ is a hyperplane, and in the
case that $\pi$ is a lower dimensional subspace.
