TITLE: Why does $L \cap (M + N) = (L \cap M) + (L \cap N) $ not hold for subspaces
QUESTION [5 upvotes]: Let $L$, $M$, and $N$ are subspaces of a vector space. Prove that following is not necessarily true.
$L \cap (M + N) = (L \cap M) + (L \cap N) $
This problem is given in 'Finite dimensional vector spaces' by Halmos. I was using 'if a vector belongs to L.H.S. then it must belong to R.H.S and vice versa' argument. Neither I can disprove it using this argument nor I could find a case where this is wrong!

REPLY [7 votes]: Try it in $\mathbb{R}^3$ with $M,N$ subspaces such that $M + N = \mathbb{R}^3$, such as $M$ being the $xy$ plane and $N$ the $z$ axis. Then $L\cap (M + N) = L$, but you should be able to find some $L$ (such as a slanted plane) for which $(L\cap M) + (L\cap N)$ is a strict subset of $L$.