I'd like to mention the following, even though it is just a reformulation: For $V\subset X $ and $I\subset X$ a closed linear subspace, the sum $V+I$ is closed in $X$ if and only if $\pi(V)$ is closed in the quotient $X/I\\ $, $\pi:X\to X/I$ being the quotient map.

(Reason: Indeed, if $\pi(V)$ is closed, so is $V+I=\pi^{-1}\pi(V)$. On the other hand, if $V+I$ is closed and $\xi\in \overline{ \pi(V) }$, then $\xi$ is limit of a sequence $\xi_n\in \pi(V) $ with $\| \xi_n - \xi_{n+1}\|_{X/I}\le 2^{-n}$ and by definition of the quotient norm there exists an inductively defined sequence $w_n\in V+I$ such that $\xi_n=\pi w_n$ and $\|w_n -w_{n+1}\| _ X < 2^{-n}$; therefore $(w_n)_n $ is a Cauchy sequence in the closed set $V + I$ and converges to an element $w\in V + I$ such that $\pi w=\xi$, which proves that $\pi(V)$ is closed in $X/I$).