Proposition 8.8.4. Let $E, F$ be metric TVSs over $K \in \RC$ with pseudonorms $\rho$ and $\eta$, respectively, and $T \in L(E; F)$. If
For any $r > 0$, there exists $C \ge 0$ such that for any $y \in T(E)$, there exits $x \in T^{-1}(y)$ with $\rho(x) \le C\eta(y)$.
$E$ is complete.
then $T(E)$ is closed.
Proof. Let $r > 0$ and $\gamma \in (0, 1)$. For any $y_{0} \in B_{F}(0, r) \cap \overline{T(E)}$, there exists $y \in B_{F}(0, r)$ such that $\eta(y) \le \eta(y_{0})$ and $\eta(y - y_{0}) \le \gamma \eta(y_{0})$. By assumption (a), there exists $x \in T^{-1}(y)$ with $\rho(x) \le C\eta(y) \le C\eta(y_{0})$.
By the method of successive approximations,
\[T(E) \supset T\braks{B_E\paren{0, \frac{Cr}{(1 - \gamma)}}}\supset B_{F}(0, r) \cap \overline{T(E)}\]
As this holds for all $r > 0$, $T(E) \supset \overline{T(E)}$.$\square$