Remark 3

Let E be a locally convex space, A an equicontinuous subset of $\mathscr{L}^f(E)$ and T a total subset of E. If $1_E$ is adherent to A for the topology of simple convergence in T, then $1_E$ is adherent to A in $\mathscr{L}_{pc}(E)$ (EVT, III, p. 16, Proposition 4 and p. 17, Proposition 5), and E is an approximation space.