We call a vector space finite-dimensional if some list of vectors in it spans the space.

For example $\mathcal{P}_m(\Z)$, the set of all polynomials with coefficients in $\Z$ and degree at most $m$, is a finite-dimensional vector space. The span of $\mathcal{P}_m(\Z)$ is $\{1, z, \dots, z^m\}$ and it is therefore finite-dimensional.