Definition 27.2.3 ($\sigma$-Derivative).label Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma \subset \mathfrak{B}(E)$ be a covering ideal, $U \subset E$ be open, $f: U \to F$, and $x_{0} \in U$, then $f$ is $\tilde \sigma$-differentiable at $x_{0}$ if there exists $V \in \cn_{E}(0)$, $T \in B_{\sigma}(E; F)$, and $r \in \mathcal{R}_{\sigma}(E; F)$ such that

for all $h \in V$.

The linear map $T \in B_{\sigma}(E; F)$ is the $\tilde \sigma$-derivative of $f$ at $x_{0}$, denoted $D_{\tilde \sigma}f(x_{0})$. If $T \in L(E; F)$, then $f$ is $\sigma$-differentiable at $x_{0}$, and $T$ is the $\sigma$-derivative of $f$ at $x_{0}$.