Is it true that $f[G+L]$ is closed in $E/M$?

Question

Let $(E, \| \cdot \|_E)$ be a Banach space, $G,L$ two closed subspaces of $E$, and $M:= G\cap L$. Then $E / M$ with its quotient norm $\| \cdot \|$ is a Banach space. Consider the quotient map $$f:E \to E/M, x \mapsto \hat x:=x +M.$$

We have $$\|\hat x\| := d(x, M) := \inf_{y\in M} \|x-y\|_E, \quad \forall x\in E.$$ Let $G+L := \{g+l \mid g\in G, l\in L\}$. Then $$f[G+L] = \{g+l+ G\cap L \mid g\in G, l\in L\}\subseteq E/M.$$

Is it true that $f[G+L]$ is closed in $E/M$?

Answer 1

The quotient map $f:E\to E/M,x\to \hat{x}:=x+M$ is continuous.

If $f[G+L]$ is closed in $ E/M $, then inverse image $G+L$ is closed in the space $(E, \| \cdot \|_E)$.

Does algebraic sum of two closed linear subspace is closed in the normed space?