Let $X$ be a Banach space.
Let $A$ and $B$ be closed subspaces of $X$.
Denote by $X/B$ the quotient space of $X$ by $B$. An element of $X/B$ is denoted by $\left[x\right]$. That is, $\left[x\right]=\left\{ x\right\} +B$.
Now, the question at hand:
Is $\left\{ \left[x\right]:x\in A\right\} $ a closed subspace of $X/B$?
(alternatively, one can consider $\text{dim}B<\infty$)