Definition 27.4.2 ($n$-Fold Differentiability).label Let $E, F$ be TVSs over $K \in \RC$ with $F$ being separated, $\sigma \subset \mathfrak{B}(E)$ be a covering ideal, 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 $\tilde \sigma$-differentiable at $x_{0}$ if

1. (1)

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

2. (2)

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

In which case, $D_{\sigma}(D_{\sigma}^{n-1}f)(x_{0}) = D_{\sigma}^{n}f(x_{0})$ is the $n$-fold $\tilde \sigma$-derivative of $f$ at $x_{0}$.

If $f: U \to F$ is $n$-fold $\tilde \sigma$-differentiable at every point in $U$, then $f$ is $n$-fold $\tilde \sigma$-differentiable on $U$. Under the identification $B_{\sigma}(E; B_{\sigma}^{n}(E; F)) = B_{\sigma}^{(n)}(E; F)$, the mapping

is the $n$-fold $\tilde \sigma$-derivative of $f$.

If for each $1 \le k \le n$, $D_{\sigma}^{k}f$ takes value in $L^{(k)}_{\sigma}(E; F)$, then $f$ is $n$-fold $\sigma$-differentiable, and $D_{\sigma}^{n}f$ is the $n$-fold $\sigma$-derivative of $f$.