Prove isomorphism 0f quotient.

Question

For $S$, $U$, $V$ being subspaces of $V$.

$V$ contains $U$, $U$ contains $S$.

How to prove: $(V/S)/(U/S)$ is isomorphic to $V/U$ ?

It's obvious when $V$ is finite. But what if $V$ is infinite ?

Answer 1

You don't consider dimension of vector spaces. It is a simple corollary of a first isomorphism theorem: it states that if $T:V\to W$ is a linear map then $$\operatorname{Im} T \cong V/\operatorname{ker}T.$$ Here $\ker T = \{v\in V : Tv=0\}.$ Take the first isomorphism theorem for the map $\pi : V/S\to V/U$ such that $\pi(v+S) = v+U$. Can you evaluate the kernel of $\pi$?