I want to prove the following proposition but I'm not sure my proof is correct. I would apreciate if someone can check if it's correct, thanks.
Let $U \subset E=E_1 \times E_2 \dots\times E_n$, where $E_i$ is a Banach space por every $i$ and $V \subset F$, $F$ a Banach space ($U,V$ open). If $f:U \rightarrow V$ is $C^1$ then $\frac{\partial f}{\partial x_i} $ is continuous.
$\textit{Proof:}$ I've already seen that $\frac{\partial f}{\partial x_i} =Df(x) \circ \iota$ where $x=(x_1,\dots, x_n)$ and $\iota$ is the canonical inclusion. So I want to see that Df(x) \circ \iota is continuous on $x$, then let $x_0 \in U$ and let's see that $\exists \delta>0$ such that $|| x-x_0|| <\delta \Rightarrow ||Df(x)\circ\iota - Df(x_0)\circ\iota || < \epsilon$
$||Df(x)\circ\iota - Df(x_0)\circ\iota || \\ = Sup ||Df(x)\circ\iota(v) - Df(x_0)\circ\iota(v) || \\ = Sup ||(Df(x)-Df(x_0))(\iota(v)) || \\ \leq Sup ||Df||\cdot||x-x_0||\cdot||\iota(v)|| \\ \leq ||Df||\cdot ||\iota||\cdot||x-x_0|| < \epsilon \\ $
The first equality is by definition of the norm on $\mathcal{L}(E,F)$, the third one is because $Df \in \mathcal{L}(E,F)$ and the last one because the only dependance on $v$ is on $\iota$ then $Sup (\iota(v)) = || \iota||$
Perhaps this is a little too late, but you can define another map $C_{\iota}: \mathcal{L}(E,F)\to \mathcal{L}(E_i, F)$ by \begin{equation} C_{\iota}(T) = T \circ \iota, \end{equation} the "composition by $\iota$ map". $C$ is a bounded (equivalently continuous) linear map, with $\rVert C \lVert \leq \rVert \iota \lVert$, because \begin{align} \lVert C_{\iota}(T) \rVert = \lVert T \circ \iota \rVert \leq \lVert T \rVert \cdot \lVert \iota \rVert, \end{align} the last inequality being true because we use the operator norm (the one with $\sup$) throughout. With this, we have that for any $\xi \in U$, \begin{align} \dfrac{\partial f}{\partial x_i}(\xi) &= Df(\xi) \circ \iota \\ &= C_{\iota}\left( Df(\xi)\right) \\ &= \left( C_{\iota} \circ Df \right)(\xi) \end{align} Since by assumption $Df: U \to \mathcal{L}(E,F)$ is continuous, and by the reasoning above, $C_{\iota}$ is also continuous, their composition is also continuous, thereby proving that $\dfrac{\partial f}{\partial x_i}(\cdot) : U \to \mathcal{L}(E_i, F)$ is continuous.