Definition 6

Let $\mathscr{T}$ be a locally convex topology on $\mathscr{L}(E)$. The $\mathscr{T}$ -resolvent topology on $\mathscr{A}(E)$ is the least fine topology such that the mappings $u\mapsto R(u, \lambda )$ of $\mathscr{A}(E)$ into $\mathscr{L}(E)$ endowed with the topology $\mathscr{T}$ are continuous for every $\lambda \in \mathbf{C}-\mathbf{R}$.