Iterated derivative

Definition

Let $U \subseteq \mathbb{R}^n$ be open and let $f \colon U \to \mathbb{R}^m$. For an integer $k \ge 0$, let $L^k(\mathbb{R}^n; \mathbb{R}^m)$ be the space of $k$-linear maps from $(\mathbb{R}^n)^k$ to $\mathbb{R}^m$, with $L^0(\mathbb{R}^n; \mathbb{R}^m) = \mathbb{R}^m$. The iterated derivatives are defined by recursion. First, $D^0 f = f$. Suppose that $D^k f \colon U \to L^k(\mathbb{R}^n; \mathbb{R}^m)$ is defined on $U$ and is differentiable on U. Then $D^{k+1} f(a)$, for $a \in U$, is the $(k+1)$-linear map $$D^{k+1} f(a)(v_1, v_2, \dots, v_{k+1}) = \big(D(D^k f)(a)(v_1)\big)(v_2, \dots, v_{k+1}).$$ Here $D(D^k f)(a)$ is the Fréchet derivative of $D^k f$ at $a$, a linear map from $\mathbb{R}^n$ to $L^k(\mathbb{R}^n; \mathbb{R}^m)$.