Space of polynomials of bounded degree

Example

Let $F$ be a field and let $n \ge 0$ be an integer. Let $F[x]_{<n}$ be the set of polynomials in $x$ with coefficients in $F$ and of degree less than $n$, with the zero polynomial included. It is a vector space over $F$, with the usual addition of polynomials and multiplication by constants. The monomials $1, x, x^2, \dots, x^{n-1}$ form a basis, so $\dim F[x]_{<n} = n$.