10.2 Linear Maps
Proposition 10.2.1. Let $E, F$ be normed vector spaces, then the topology on $L_{b}(E; F)$ is induced by the operator norm
\[\norm{\cdot}_{L(E; F)}: L(E; F) \to [0, \infty) \quad T \mapsto \sup_{\substack{x \in E \\ \norm{x}_E = 1}}Tx\]
Proof. By Proposition 9.7.1.$\square$