I'm wondering if the 'Normed Vector Space version' of the following theorem holds:
Theorem: let $A\subseteq \mathbb{R}^n$ be open and let $f:A\to \mathbb{R}^m$ have continuous partial derivatives $\partial f_i/\partial x_j$ on $A$. Then $f$ is differentiable on $A$.
In particular, the total derivative can be extended to NVS by considering the Fréchet Derivative. If we extend the definition of partial derivatives as follows:
Throughout the post let $V$ and $W$ be NVSs, with $V'\subseteq V$ open and $f$ a function $V'\to W$.
Definition: given $a\in V'$ and a vector $v\in V$, we define (assuming the limit exists) $$\frac{\partial f}{\partial v}(a) := \lim_{t\rightarrow 0}\frac{f(a+tv)-f(a)}{t},$$
and, in the case $v=e_i$ for a basis vector $e_i$,
$$\frac{\partial f}{\partial x_i}(a) := \frac{\partial f}{\partial e_i}(a).$$
then, is either of the following statements true?
Theorem (?): if $f:V'\to W$ has continuous partial derivatives $\partial f/\partial v$ (for any $v\in V$) on $A$, then $f$ is differentiable on $A$.
Theorem (?): let $\dim V = n$ and $\dim W = m$. If $f:V'\to W$ have continuous partial derivatives $\partial f_i/\partial x_j$ on $A$, then $f$ is differentiable on $A$.