Definition 18.2.3 (Derivative). Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma \subset B(E)$ be an upward-directed family that contains all finite sets, $U \subset E$ be open, $f: U \to F$, and $x_{0} \in U$, then $f$ is $\sigma$-differentiable at $x_{0}$ if there exists $V \in \cn_{E}(0)$, $T \in L(E; F)$, and $r \in \mathcal{R}_{\sigma}(E; F)$ such that
\[f(x_{0} + h) = f(x_{0}) + Th + r(h)\]
for all $h \in V$.
The linear map $T \in L(E; F)$ is the $\sigma$-derivative of $f$ at $x_{0}$, denoted $D_{\sigma}f(x_{0})$.