It is known that if $U_1,...,U_n$ are subspaces of $V$ then $U_1 \oplus\cdots\oplus U_N = V $ iff $\dim V = \dim U_1 + \cdots + \dim U_n$ and $V = U_1+\cdots+U_n$
But $$\dim(U_1+U_2+U_3) = \dim(U_1)+\dim(U_2)+\dim(U_3)-\dim(U_1 \cap U_2)-\dim(U_1 \cap U_3)-\dim(U_2 \cap U_3)+\dim(U_1 \cap U_2 \cap U_3)$$ does not always hold.
Is there a condition on $U_1,...,U_n$ this generalized equality would imply? For example, consider in $\mathbb{R}^3$, $U_1 = xy$-plane, $U_2 = yz$-plane, $U_3 = xz$-plane satisfy the formula for $V=\mathbb{R}^3$.
