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. 

Define the map $T\colon V\rightarrow \mathbb{R}^4$ by
$$T\left(\sum_{i=0}^n a_ix^i\right) = (a_0,a_1,a_2,a_3),$$
meaning that you keep only the first four coefficients of any polynomial.
This is a surjective linear map, and its kernel is
$$\text{ker}(T) = \left\{\sum_{i=4}^n a_ix^i\ |\ n\in\mathbb{N}, a_i\in\mathbb{R}\right\}.$$
Note that $\text{ker}(T) = W$. The induced map
$$\tilde{T}\colon V/W\rightarrow\mathbb{R}^4$$
is thus an isomorphism.
