Definition 18.4.1 ($n$-Fold Differentiability). 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, $\mathcal{H}\subset B_{\sigma}(E; F)$ be a subspace, and $\mathcal{R}_{\sigma} = \mathcal{R}_{\sigma}(E; F)$.

Let $U \subset E$ be open, $f: U \to F$, $x_{0} \in U$, and $n > 1$, then $f$ is $n$-fold $\sigma$-differentiable at $x_{0}$ if

1. There exists $V \in \cn_{E}(x_{0})$ such that $f$ is $(n-1)$-fold differentiable on $V$.

2. The derivative $D_{\sigma}^{n-1}f: U \to B^{n-1}_{\sigma}(E; F)$ is derivative at $x_{0}$.

In which case, $D_{\sigma}(D_{\sigma}^{n-1}f)(x_{0}) \in L(E; B^{n-1}_{\sigma}(E; F))$ is the $n$-fold $\sigma$-derivative of $f$ at $x_{0}$.

The mapping $f: U \to F$ is $n$-fold $\sigma$-differentiable on $U$ if it is $n$-fold $\sigma$-differentiable at every point in $U$. Under the identification $B_{\sigma}(E; B^{n-1}_{\sigma}(E; F)) = B_{\sigma}^{n}(E; F)$ given by Proposition 8.11.3,

is the $n$-fold $\sigma$-derivative of $f$ at $x_{0}$.