Quotient Spaces of Infinite Dimension Let $V$ be a vector space of polynomials over $\mathbb{R}$ and let
$W$ be a subspace of polynomials divisible by $x^4$,
find the dimension of $V/W$. I can solve if $V$ is finite dimensional but don't know what to do if $V$ is infinite dimensional. 

Actually, $V$ is a $k$-algebra, and $W$ is an ideal of this algebra. As a quotient algebra $V/W$ is $k[x]/(x^4)$, isomorphic as a vector space to $\{f(x)\in k[x]\mid \deg f(x)\le 3\}$.
Thus a basis for $V/W$ is $(1+W, x+W, x^2+W, x^3+W)$.
