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