We call a [[vector space]] **finite-dimensional** if some *finite* list of vectors in it [[linear span]]s the space. - eg the space of [[polynomial]]s of degree $m \in \mathbb{N}$ is finite-dimensional since it is spanned by the [[monomial]]s $1, z, \cdots, z^m$. Otherwise we call it **infinite-dimensional**. e.g. the [[function space]] of [[continuous]] functions. all finite-dimensional spaces are [[isomorphism|isomorphic]] to $\mathbb{F}^{N}$ and thus to each other via [[coordinate]]s # sources [[2015AxlerLinearAlgebraDone|Axler]] 2.10, 2.17