Dense open-unit-ball image criterion
= Dense open-unit-ball image criterion
Let $T:X\to X$ be a <bounded linear operator> on a <Banach space>, and let $B$ be the open unit ball. If $B\subseteq\overline{T(B)}$, then $B\subseteq T(B)$. Scale the density statement and correct a target by a geometrically decreasing sequence of approximate preimages. Completeness makes their sum converge inside $B$, and continuity of $T$ makes its image equal the target.