Quotient operator ($B_Y(0,c)\subseteq T(B_X(0,1))$)

A quotient operator is a surjective bounded linear operator that is an open map. It is equivalent to requiring a ball about zero in the image of the open unit ball. Translation and scaling then prove being an open map at every point. For a Banach space domain, a bounded linear operator is a quotient operator exactly when its Banach-space adjoint is an isomorphic embedding: a lower bound on the adjoint gives approximate ball-surjectivity by Hahn-Banach separation, and successive approximation for quotient operators gives exact preimages.

Let $T:X\to Y$ be a bounded linear operator with $X$ a Banach space. Suppose every $\Vert y\Vert <1$ has an approximation $Tx$ with $\Vert x\Vert <1$ and error less than $\delta \in (0,1)$. Repeatedly approximate the scaled residual to construct $x_n$ with $\Vert x_n\Vert <{\delta }^n$ and residual less than ${\delta }^{n+1}$. Completeness sums the series, and continuity gives $T\sum x_n=y$. Since $\Vert \sum x_n\Vert <(1-\delta {)}^{-1}$, scaling proves $B_Y(0,1-\delta )\subseteq T(B_X(0,1))$. Hence $T$ is a quotient operator, even without assuming completeness of $Y$.