Definition 18.2.4 (Differentiable). 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, and $f: U \to F$, then $f$ is $\sigma$-differentiable on $U$ if it is $\sigma$-differentiable at every point in $U$. In which case, the map $D_{\sigma} f: U \to L(E; F)$ is the $\sigma$-derivative of $f$.