In Kolk's Multidimensional Real Analysis I: Differentiation
He used the following Corollary 2.4.3
to prove that a multilinear(k-linear) map $T \in Lin^k(\mathbf{R}^n,\mathbf{R})$ is differentiable.(In the proof of proposition 2.7.6)
The whole argument is a single sentence:
The differentiability of T follows from the fact that $T(a_1,\cdots, a_k) \in \mathbf{R}$ are polynomials in the coordinates of vectors $a_1,\cdots, a_k \in \mathbf{R}^n$. See corollary 2.4.3.
I know the fact that $T(a_1,\cdots, a_k) \in \mathbf{R}$ are polynomials in the coordinates of vectors $a_1,\cdots, a_k \in \mathbf{R}^n$. It's a basic property of multilinear map. But how does the conclusion relate to corollary 2.4.3?
Note that he defines the product $\langle f_1, f_2 \rangle$ as the composition:
$ \begin{equation} \langle f_1, f_2 \rangle: x \mapsto (f_1(x),f_2(x)) \mapsto \langle f_1(x), f_2(x) \rangle : \mathbf{R}^n \rightarrow \mathbf{R}^p \times \mathbf{R}^p \rightarrow \mathbf{R} \end{equation} $
If we write $$a_{i}=\sum_{\ell=1}^{n}b_{i,\ell}e_{\ell},$$ then $$T(a_{1},\ldots,a_{k})=T \left( \sum_{j_{1}=1}^{n}b_{1,j_{1}}e_{j_{1}},\ldots \sum_{j_{k}=1}^{n} b_{k,j_{k}}e_{j_{k}}\right)=\sum_{j_{1}=1}^{n}\cdots \sum_{j_{k}=1}^{n}T(e_{j_{1}},\ldots,e_{j_{k}})b_{1,j_{1}}\cdots b_{k,j_{k}}.$$
By part (i) of Corillary 2.4.3, its is enough to show that $S(a_{1},\ldots,a_{k})=b_{1,j_{1}}\cdots b_{k,j_{k}}$ is differentiable. You can think $S$ as the product of the maps of the form \begin{align*} S_{1,j_{1}}:\mathbb{R}^{n} \times \cdots \times \mathbb{R}^{n} & \to \mathbb{R} \\ (a_{1},\ldots,a_{k}) & \mapsto b_{1,j_{1}}. \end{align*} Thus, $S$ is differentiable using (ii) $k$-times.