Is there a (valid) formula for $\dim(U + V + W)$? I know from MO that $\begin{align*} \dim(U + V + W) &= \dim(U) + \dim(V) + \dim(W)\\ &\qquad\mathop{-} \dim(U \cap V) - \dim(U \cap W) - \dim(V \cap W)\\&\qquad \mathop{+} \dim(U \cap V \cap W) \end{align*}$ is wrong.
Can we relate $\dim(U + V + W)$ with the cardinality of some of their quotient spaces? (sorry if this is a dummy question but I'm not any familiar with quotient spaces).