Example 2

Let $\mathscr{T}_b$ be the Banach-space topology of $\mathscr{L}(E)$. For the $\mathscr{T}_b$-resolvent topology, every $u\in \mathscr{A}(E)$ admits a countable fundamental system of neighbourhoods.

In fact, it is enough to prove that for every $\lambda \in$ C-R and every $\varepsilon >0$, there exist $\lambda '\in \mathbf{Q}+i\mathbf{Q}^*$ and an integer $n\geqslant 1$ such that every partial operator $v\in \mathscr{A}(E)$ satisfying $\|R(v, \lambda ')-R(u, \lambda ')\|<1/n$ also satisfies the condition $\|R(v, \lambda )-R(u, \lambda )\|< \varepsilon$; writing

this property follows from formula (8) of IV, p. 245, from the estimates of proposition 17 of IV, p. 248, and from the fact that $\mathbf{Q}+i\mathbf{Q}^*$ is everywhere dense in $\mathbf{C}-\mathbf{R}$.

The conclusion of the proposition is not valid in general if $\mathscr{C}_0(\mathbf{R})$ is replaced by $\mathscr{C}_b(\mathbf{R})$ (exercise 29 of IV, p. 366, e)). Nevertheless, one has the following result.