Quotient Space Manipulations

Question

Let $A,B,C$ be subspaces of $V$. Given that $(A+B)/(A\cap B)$ and $(B+C)/(B\cap C)$ is finite dimensional, is it true that $(A)/(A\cap B\cap C)$ is finite dimensional?

Answer 1

Probably you know this terminology, but just in case: we say a vector subspace $U$ is of finite codimension in a vector space $W$ if $W/U$ is finite dimensional. Let's use the notation $U<W$ to denote "$U$ is a subspace of $W$", and $U\ll W$ to denote "$U$ is a subspace of finite codimension in $W$". Note that being of finite codimension is preserved by intersection; that is, if $U,V$ are subspaces of $W$ of finite codimension, then so is $U\cap V$. (This follows from the standard isomorphism theorems: $(U+V)/V\cong U/U\cap V$.) Also note that if $U\ll W$ and $U<V<W$, then $U\ll V\ll W$.

Then we can note that $A\cap B \ll A+B$, and therefore $A\cap B\ll A$ and $A\cap B\ll B$. Likewise, $B\cap C \ll B$. Then using our intersection property, $A\cap B\cap C=(A\cap B)\cap(B\cap C)\ll B$. But in particular, $A\cap B\cap C\ll A\cap B\ll A$ hence $A\cap B \cap C$ is of finite codimension in $A$, which means $A/(A\cap B\cap C)$ is finite dimensional.