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Three subspaces $L$, $M$, and $N$ of a vector space $V$ are called independent if each one is disjoint from the sum of the other two. Prove that a necessary and sufficient condition for $V=L\bigoplus(M\bigoplus N)$ (and also for $V=(L\bigoplus M)\bigoplus N$) is that $L$, $M$, and $N$ be independent and that $V=L+M+N$.

I am aware that necessary and sufficient is the same as if and only if. I'm confused by the question, partially because when it says "disjoint from the sum of the other two" is it talking about the direct sum or the spanning sum? I think I need some help getting the proof started as well. Thank you in advance.

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    "Disjoint" seems impossible, since all three subspaces contain the zero vector. Sloppy writing on the part of the question's author. – 2017-02-04
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    Hint construct bases in each subspace and prove they form a basis for the whole space. – 2017-02-06

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