Finite-dimensional vector space

Definition

Let $F$ be a field and let $V$ be a vector space over $F$. The space $V$ is finite-dimensional when it has a finite spanning set. Otherwise $V$ is infinite-dimensional.

Remarks

A finite-dimensional space has a finite basis, because every spanning set contains a basis. The spaces $F^n$ and $M_{m \times n}(F)$ are finite-dimensional. The polynomial ring $F[x]$ is infinite-dimensional: finitely many polynomials have a bounded degree, so they do not span $F[x]$. Each subspace of a finite-dimensional space is finite-dimensional.

Explanations

L1 human

A space is finite-dimensional when finitely many vectors are enough to build every vector. The plane needs two vectors, and ordinary space needs three. The space of all polynomials needs infinitely many, because the degrees have no bound.