Proposition 18.2.2. Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma \subset B(E)$ be an upward-directed family contains all finite sets, $B_{\sigma}(E; F)$ be the space of linear operators bounded on sets in $\sigma$, and $\mathcal{R}_{\sigma}(E; F)$ be the space of $\sigma$-small functions, then $(B_{\sigma}(E; F), \mathcal{R}_{\sigma}(E; F))$ is a system of derivatives and remainders.
Proof. Let $T \in B_{\sigma}(E; F)$ and suppose that there exists $V \in \cn_{E}(0)$ circled and $r \in \mathcal{R}_{\sigma}(E; F)$ such that $T|_{V} = r|_{V}$. For any $x \in V$, $\bracs{x}\in \sigma$, so $T(x) = \lim_{t \downto 0}T(tx)/t = 0$ as $F$ is separated.$\square$