Banach-Schauder Theorem/Lemma 2

Lemma
Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be Banach spaces.

Let $T : X \to Y$ be a surjective bounded linear transformation.

Let $r > 0$ be such that:


 * $\map {B_Y} {0, r} \subseteq \paren {\map T {\map {B_X} {0, m} } }^-$

where:
 * $\map {B_Y} {0, r}$ denotes the open ball in $Y$ centered at $0 \in Y$ with radius $r$
 * $\map {B_X} {0, m}$ denotes the open ball in $X$ centered at $0 \in X$ with radius $m$
 * $\paren {\map T {\map {B_X} {0, m} } }^-$ denotes the topological closure of $\map T {\map {B_X} {0, m} }$.

Then:
 * $\map {B_Y} {0, r} \subseteq \map T {\map {B_X} {0, 2 m} }$