I am wondering about a problem from the calculus of derivatives in Banachspaces. It is about the difference between two definitions of continuity concerning the Gâteaux-Derivative of a function $P:X\rightarrow Y$ in Banach spaces $X,Y$.The Gâteaux-Derivative in $f$ in direction $h$ is then defined as: \begin{align} DP(f)h=lim_{t \rightarrow 0}[P(f+th)-P(f)]/t \end{align} There are now two different ways in defining the continuity of $DP$. The first way is that $DP$ as a function in the two variables $f$ and $h$ is continuous on the Product-Space: \begin{equation} DP:(U\subset X)\times X \rightarrow Y \ \ \ \ \ (1) \end{equation} is continuous. The second way is two consider $DP$ as a map from the Banach space $X$ or better a subset $U \subset X$ to the space $L(X,Y)$ of the linear functionals from $X$ to $Y$: \begin{equation} DP:U\rightarrow L(X,Y) \ \ \ \ \ \ \ \ \ \ \ \ \ (2) \end{equation} is continuous. The question now is if these two definitions are equivalent in the case of Banach spaces. In the case of Fréchet-Spaces $X,Y$ its obviously not equivalent because in general $L(X;Y)$ is not a Fréchet-Space anymore, its not metrizable in general . But what about the Banach spaces? Is it equivalent in this case? What we know is that when $P$ is Gâteaux-differentiable and $DP$ continuous,in the sense of continuity on product spaces, then $P$ is also Fréchet-differentiable and the Fréchet-Derivative is the same as the Gâteaux-Derivative. As discussed for example in Gâteaux derivative. But is it also continuous in the sense of continuity of Fréchet-Derivatives, continuous as a map described in (2). Does anybody maybe know a counterexample?
I think that an example for a function, gateux-differentiable and continous in the first sense on the product space $U\times Y$, but not continuous as a function $U\rightarrow L(X,Y)$ should be the function $P:L^{1}([0,\pi])\rightarrow \mathbb{R}$ given by \begin{equation} f\rightarrow \int\limits_{0}^{\pi} \sin(f(t))dt \end{equation} This function is described in, http://www.m-hikari.com/ams/ams-password-2009/ams-password17-20-2009/gaxiolaAMS17-20-2009.pdf, where it is shown that $P$ is gateux-differentiable but not frechet-differentiable. It is not continuous in the second sense, which one can easily see by testing with functions centered on a small Intervall with values $n \in \mathbf{N}$. But it should be continuous on the product space $U\times X$.