TITLE: Examples of bounded linear operators with range not closed
QUESTION [13 upvotes]: I've been trying to get some intuition on what it means for a bounded linear operator to have closed range. Can anyone give some simple examples of such an operator that does not have closed range?
Thanks!

REPLY [6 votes]: Take the bounded linear operator $$T: l^\infty \ni (x_j) \mapsto \left(\dfrac{x_j}{j}\right)\in l^\infty.$$
Then $x^{(n)}=(\sqrt{1},\sqrt{2},\dots,\sqrt{n},0,0,\dots)$ is in $l^\infty$ with $Tx^{(n)}=(1/\sqrt{1},\dots,1/\sqrt{n},0,\dots)$.
We have $Tx^{(n)} \to y=(1/\sqrt{j})$ in $l^\infty$, but $y \notin T(l^\infty)$ as $(\sqrt{j})\notin l^\infty$. So $T(l^\infty)$ is not closed.