Speaking of the relation between Gâteaux and Fréchet, authors usually point out that $$\text{Fréchet} \implies \text{Gâteaux}$$ and then give a counterexample to illustrate that the converse doesn't hold.
Say, we have a function which is known to be Gâteaux differentiable (all directional derivatives exist). I wonder what conditions need to be added in order to insure Fréchet differentiability?
In case the function goes from $\mathbb R^n$, the continuity of partial derivatives does the job. But what if the function is between two Banach spaces? Would it suffice if all directional derivatives are continuous?
Update: I have found the following theorem, unfortunately without any proof.
Let $X,Y$ be Banach spaces and $U\subset X$ open. If the following conditions hold:
- $f:U\to Y$ has continuous Gâteaux derivative \begin{align} df: U \times X & \to Y \\ (x,h) & \mapsto df(x,h) \end{align}
- the linear operator $df(x,\cdot)$ is bounded for all $x\in U$
- the function \begin{align}U & \to \mathcal{L}(X,Y) \\ x & \mapsto df(x,\cdot) \end{align} is continuous
then $f\in C^1(U,Y)$.
Does it look reasonable? Where could I find the proof?