Let $X$ be open in $\mathbb R^n$, $F$ a Banach space, and $m \in \mathbb N^*$. Suppose $f:X \to F$ such that $\partial_{j_1} \cdots \partial_{j_{m+1}} f$ and $\partial^m f$ exist in a neighborhood of $a$ for all $j_1, \ldots, j_{m+1} \in \{1,\ldots,n\}$. Assume $h^i = \left (h_1^i, \ldots, h_n^i\right ) \in \mathbb R^n$ with $1 \le i \le m$. We define a map $A$ by $$\begin{array}{l|rcl} A & {(\mathbb R^n)}^m & \longrightarrow & F \\ & \left [h^1, \ldots,h^m\right ] & \longmapsto & \sum_{j_1, \ldots, j_m =1}^n \partial_j \partial_{j_1} \cdots \partial_{j_m} f (a) \left (h^1_{j_1} \cdots h^m_{j_m}\right ) \end{array}$$
I have verified that $A$ is a multilinear map and thus $A \in \mathcal L^m(\mathbb R^n, F)$. Moreover, it follows from the definition of mixed partial derivative that $\partial^m f(a) \in L^m(\mathbb R^n, F)$. In the following, I try to prove that $$\partial_j (\partial^m f)(a) = A$$
I'm not sure if, in (5), I correctly move the lim operator inside the norm operator. Could you please verify if this step is correct? Thank you so much!
My attempt:
First, we have $$\begin{aligned} &\frac{\partial^m f(a +te_j) - \partial^m f(a)}{t} \left [h^1, \ldots,h^m\right ] - A \left [h^1, \ldots,h^m\right ] \\ ={}& \frac{\partial^m f(a +te_j)[h^1, \ldots,h^m] - \partial^m f(a) [h^1, \ldots,h^m]}{t} - A [h^1, \ldots,h^m]\\ ={}& \frac{ \sum_{j_1, \ldots, j_m =1}^n \partial_{j_1} \cdots \partial_{j_m} f (a + te_j) \left (h^1_{j_1} \cdots h^m_{j_m}\right )- \sum_{j_1, \ldots, j_m =1}^n \partial_{j_1} \cdots \partial_{j_m} f (a) \left (h^1_{j_1} \cdots h^m_{j_m}\right )}{t} \\ & \quad - \sum_{j_1, \ldots, j_m =1}^n \partial_j \partial_{j_1} \cdots \partial_{j_m} f (a) \left (h^1_{j_1} \cdots h^m_{j_m}\right ) \\ ={}& \sum_{j_1, \ldots, j_m =1}^n \left (h^1_{j_1} \cdots h^m_{j_m} \right) \cdot \left ( \frac{ \partial_{j_1} \cdots \partial_{j_m} f (a + te_j) -\partial_{j_1} \cdots \partial_{j_m} f (a) }{t} - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (a) \right ) \end{aligned}$$
It follows that
$$\begin{aligned} & \lim_{t \to 0} \left \| \frac{\partial^m f(a +te_j) - \partial^m f(a)}{t} - A\right \| \\ \overset{(1)}{=}{}& \lim_{t \to 0} \sup_{\|h^1\|_1\le1,\ldots,\|h^m\|_1 \le 1} \left \| \sum_{j_1, \ldots, j_m =1}^n \frac{\partial^m f(a +te_j) - \partial^m f(a)}{t} \left [h^1, \ldots,h^m\right ] - A \left [h^1, \ldots,h^m\right ] \right \|\\ \overset{(2)}{\le}{}& \lim_{t \to 0} \sup_{\|h^1\|_1\le 1,\ldots,\|h^m\|_1 \le 1} \sum_{j_1, \ldots, j_m =1}^n \left |h^1_{j_1} \cdots h^m_{j_m} \right | \cdot\bigg \| \frac{ \partial_{j_1} \cdots \partial_{j_m} f (a + te_j) - \partial_{j_1} \cdots \partial_{j_m} f (a) }{t} \\ \overset{(3)}{\le}{}& \lim_{t \to 0} \sup_{\|h^1\|_1\le 1,\ldots,\|h^m\|_1 \le 1} \sum_{j_1, \ldots, j_m =1}^n \|h^1\|_1 \cdots \|h^m\|_1 \cdot\bigg \| \frac{ \partial_{j_1} \cdots \partial_{j_m} f (a + te_j) - \partial_{j_1} \cdots \partial_{j_m} f (a) }{t} \\ & \quad - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (a)\bigg \|\\ \overset{(4)}{\le}{}& \lim_{t \to 0} \sum_{j_1, \ldots, j_m =1}^n \left \| \frac{ \partial_{j_1} \cdots \partial_{j_m} f (a + te_j) - \partial_{j_1} \cdots \partial_{j_m} f (a) }{t} - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (a)\right \|\\ ={}& \color{blue}{\sum_{j_1, \ldots, j_m =1}^n \lim_{t \to 0} \left \| \frac{ \partial_{j_1} \cdots \partial_{j_m} f (a + te_j) - \partial_{j_1} \cdots \partial_{j_m} f (a) }{t} - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (a)\right \|}\\ \overset{(5)}{=}{}& \color{blue}{\sum_{j_1, \ldots, j_m =1}^n \left \| \lim_{t \to 0} \frac{ \partial_{j_1} \cdots \partial_{j_m} f (a + te_j) - \partial_{j_1} \cdots \partial_{j_m} f (a) }{t} - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (a)\right \|}\\ ={}& \sum_{j_1, \ldots, j_m =1}^n \left \| \partial_j \left ( \partial_{j_1} \cdots \partial_{j_m} f \right ) (a) - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (a) \right \|\\ ={}& \sum_{j_1, \ldots, j_m =1}^n \left \| \partial_j \partial_{j_1} \cdots \partial_{j_m} f (a) - \partial_j \partial_{j_1} \cdots \partial_{j_m} f (a) \right \|\\ ={}& 0 \end{aligned}$$
$(1)$: This follows from the definition of the operator norm of a multilinear map.
$(2)$: This follows from triangle inequality.
$(3)$: It follows from the definition of $\|\cdot\|_1$ that $|h^1_j| \le \|h^1\|_1,\ldots, |h^m_j| \le \|h^m\|_1$ for all $j \in \{1,\ldots,n\}$. As such, $\left |h^1_{j_1} \cdots h^m_{j_m} \right | =\left |h^1_{j_1}\right | \cdots \left | h^m_{j_m} \right | \le \|h^1\|_1 \cdots \|h^m\|_1$.
$(4)$: It follows from $\|h^1\|_1\le 1,\ldots,\|h^m\|_1 \le 1$ that $\|h^1\|_1 \cdots \|h^m\|_1 \le 1$.
Hence $$\partial_j (\partial^m f)(a) = A$$
