The left support $l(x)$ of an operator $x$ between Hilbert spaces $\mathbb{H}$ and $\mathbb{K}$ is defined as the smallest projection $e \in \mathfrak{B}(\mathbb{H})$ such that $ex=x$.
The question is to prove that range of $l(x)=$cl{range $x$}. I was able to prove that cl{range $x$}$\subseteq $range of $l(x)$ but not conversely.
Hint: The projection operator $e:=\mathbf{P}_{\overline{\operatorname{Ran}(x)}}$ satisfies the equality $ex=x$.