It disturbs me a lot how to do with "Infinite" space in solving problems like the following one.
Let S$_{1}$, S$_{2}$ and V$_{1}$, V$_{2}$ be subspaces of V. Suppose that V=V$_{1}$$\oplus$V$_{2}$ and S=S$_{1}$$\oplus$S$_{2}$.
$\oplus$ means direct sum here. And S$_{1}$$\subset$V$_{1}$ ,S$_{2}$$\subset$V$_{2}$.
Prove that V/S=(V$_{1}$$\oplus$V$_{2}$)/(S$_{1}$$\oplus$S$_{2}$) is isomorphic to (V$_{1}$/S$_{1}$)$\times$(V$_{2}$/S$_{2}$).
Define:
$$\phi:V\to V_1/S_1\times V_2/S_2\;,\;\;\phi v:=\left(v_1+S_1,\,v_2+S_2\right)$$
where we've written $\;v=v_1+v_2\;,\;\;v_i\in V_i\;$, and this expression is unique (why?) .
Show $\;\phi\;$ is a linear map and also that $\;\ker V=S=S_1\oplus S_2\;$