TITLE: The calculation of $\dim(U + V + W)$
QUESTION [9 upvotes]: 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).

REPLY [8 votes]: I don't think you can do better than this:
$$\begin{align*} \dim(U +V + W) &= \dim((U +V) + W) \\ &= \dim(U +V) + \dim W - \dim((U+V)\cap W) \\ &= 
\dim U + \dim V - \dim (U \cap V) + \dim W - \dim((U+V)\cap W) \end{align*}$$
Now you're stuck with $\dim((U+V)\cap W) $, for which there does not seem to be a simple formula.
(BTW, see also https://mathoverflow.net/questions/17740/is-there-a-version-of-inclusion-exclusion-for-vector-spaces but you probably know about this already.)