# Is $GL(E)$ dense in $L(E)$, when $\dim E=\infty$?

Let $E$ be a normed vector space (Banach space, if you like).

Is $GL(E)$, the set of invertible and continuous endomorphism of $E$, dense in $L(E)$, the set of continuous endomorphism of $E$?

I specify that I know the answer if $dim(E)<\infty$, with classical arguments about the spectrum of matrices, and, I know that $GL(E)$ is open in $L(E)$, even if $dim(E)=\infty$ (if $E$ is a Banach space), using the formula $(I-u)^{-1}=\sum_{n\in\mathbb{N}}u^n$ for $u$ small enough.

So the remaining question I would like to ask is about the density of $GL(E)$ in $L(E)$, and in the case it is not, about its closure.

• Searching a bit results in: math.stackexchange.com/questions/92079/… – Mariano Suárez-Álvarez Jan 10 '16 at 22:21
• Thanks, couldn't find it, though I assure you I tried... (sorry, a bit new to this, I guess I don't master the research process yet) – John Steinbeck Jan 11 '16 at 23:32
• Use google (and add «site:math.stackexchange.com» to your search to restrict results to ones in this site) – Mariano Suárez-Álvarez Jan 11 '16 at 23:38