Originally Posted by

**johng** Presumably, this question arose from the true statement that $\displaystyle dim(U_1+U_2)=dim(U_1)+dim(U_2)-dim(U_1\cap U_2)$. When you try to extend this to 3 subspaces, you need to find $\displaystyle (U_1+U_2)\cap U_3$, which is definitely not equal to $\displaystyle (U_1\cap U_3)+(U_2\cap U_3)$. Basically, this is why Deveno's counter example above works. Notice though if $\displaystyle U_1\subset U_3$ or $\displaystyle U_2\subset U_3$, then $\displaystyle (U_1+U_2)\cap U_3=(U_1\cap U_3)+(U_2\cap U_3)$, and so in this case your original statement is true. (It boils down to the true statement about two subspaces.)