Proposition 10.5.3.label Let $E, F$ be TVSs over $K \in \RC$ and $T \in L(E; F)$, then for any $B \subset E$ bounded, $T(B)$ is also bounded.
Proof. Let $U \in \cn_{F}(0)$, then $T^{-1}(U) \in \cn_{E}(0)$, so there exists $\lambda \in K$ such that $\lambda T^{-1}(U) = T^{-1}(\lambda U) \supset B$ and $\lambda U \supset T(B)$.$\square$