# Why does $L \cap (M + N) = (L \cap M) + (L \cap N)$ not hold for subspaces

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!

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A related post on MathOverflow: mathoverflow.net/questions/23478/… – Jonas Meyer Mar 21 '11 at 22:42

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 $L$.
The same works in $\mathbb{R}^2$ with $M,N$ the axes and $L$ a slanted line. The RHS intersections are both the origin. – Ross Millikan Mar 21 '11 at 14:09
@Ross: Thanks, mine was just the first example I thought of. However, why did you replace $L$ with $\mathbb{R}^3$ at the end of my answer? $(L\cap M) + (L\cap N)$ would be a strict subset of $L$, and just being a strict subset of $\mathbb{R}^3$ would not make the OP's statement in question false. – Alex Becker Mar 21 '11 at 21:56