Proposition 12.3.1.label 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 11.10.1.$\square$
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