TITLE: Distributivity of subspaces
QUESTION [6 upvotes]: I need to either give a proof or find a counterexample to a statement:
$$L+(M∩N) = (L + M)∩(L + N)$$
Where $L$,$M$,$N$ are subspaces of a vector space $V$. 
I could do $LHS⊆RHS$ proof, but I'm stuck with backwards proof. I would be really grateful if someone could help me out.

REPLY [12 votes]: Take $M=\mathbb R\times\{0\}$ and $N=\{0\}\times \mathbb R$. Then set $L=\{(x,x)|x\in\mathbb R\}
$

Then, $M\cap N=\{(0,0)\}$ (should be obvious why) meaning that the left side is equal to $L+\{0\}=L$. 
But $L+M = L+N = \mathbb R^2$ (this should be easy to prove), meaning the right side is equal to  $\mathbb R^2\cap \mathbb R^2 = \mathbb R^2$.

Since $L\neq \mathbb R^2$, clearly, the equality does not hold, and you can only conclude that $LHS\subseteq RHS$.