is the intersection of finite codimensional subspaces of finite codimension in an infinite dim vector space?
Let $X$ be an infinite dimensional vector space, with subspaces $V,W$ such that $\dim X/V,\dim X/W<\infty$. I want to know if $\dim X/(V\cap W)<\infty$. I want to say it is true, but am having trouble proving this.
Best Answer
Sure, $(V+W)/W\cong V/(V\cap W)$, that’s one of the Isomorphism Theorems. Since the left-hand side is finite-dimensional, so is the right. Then you have the inclusions $V\cap W\subset V\subset X$. Both the quotients here are finite-dimensional, so the all-the-way quotient $X/(V\cap W)$ is finite-dimensional.