Let $X$ be a Banach space and $V\subset X$ a subspace. Suppose that the quotient space $X/V$ is finite dimensional. Is $V$ then closed?
In other words: if $V$ is not closed, then can $X/V$ be finite dimensional?
(I think the answer is NO)
Let $V$ be the kernel of a discontinuous linear functional $f$. Then $X/V$ is one-dimensional but $V$ isn't closed (open mapping theorem).