Let $T: E \to F$ be a linear continuous function between Banach spaces. Let $$ B_E (x, 1) := \{z\in E \mid |z-x| <1\} \quad \text{and} \quad B_F (y, 1) := \{z\in F \mid |z-y| <1\} \quad \forall x \in E, \forall y \in F. $$
Previously, I proved that
Open Mapping Theorem: The following statements are equivalent:
- $T$ is surjective.
- There is $r>0$ such that $B_F (Tx, r) \subseteq T [B_E (x, 1)]$ for all $x \in E$.
- There is $r>0$ such that $B_F (Tx, r) \subseteq \overline{T [B_E (x, 1)]}$ for all $x \in E$.
Now I want to generalize above OMT a little bit, i.e.,
Corollary: The following statements are equivalent:
- $T$ is surjective.
- There is $r>0$ such that $B_F (Tx, r r') \subseteq T [B_E (x, r')]$ for all $x \in E$ and for all $r'>0$.
- There is $r>0$ such that $B_F (Tx, rr') \subseteq \overline{T [B_E (x, r')]}$ for all $x \in E$ and for all $r'>0$.
Could you have a check on my attempt?
My attempt: Assume $T$ is surjective.
By MPT, there is $r>0$ such that $B_F (T \frac{x}{r'}, r) \subseteq T [B_E (\frac{x}{r'}, 1)]$ for all $x \in E$. Equivalently, $$ T \frac{x}{r'} + B_F (0, r) \subset T \left [ \frac{x}{r'} + B(0, 1) \right ]. $$ It follows that $$ T x + B_F (0, rr') \subset T \left [ x + B(0, r') \right ]. $$ The claim then follows.
By MPT, there is $r>0$ such that $B_F (T \frac{x}{r'}, r) \subseteq \overline{T [B_E (\frac{x}{r'}, 1)]}$ for all $x \in E$. Equivalently, $$ T \frac{x}{r'} + B_F (0, r) \subset \overline{T \left [ \frac{x}{r'} + B(0, 1) \right ]}. $$ Notice that $\alpha \overline A = \overline{\alpha A}$, so $$ T x + B_F (0, rr') \subset \overline{T \left [ x + B(0, r') \right ]}. $$ The claim then follows.
