Suppose I have the operator
$$T:L_2[0,1]\to L_2[0,1],$$ $$T[x(t)] = \sin(x(t)).$$ My first question: Is it true that the Gateaux derivative of this operator equals to the ordinary derivative of $\sin(x(t))$? And how can I show that this operator does not have a Fréchet derivative anywhere?
Let's take some $h \in L_2[0,1]$ and compute the directional derivative with respect to $h.$ Then we get something like
$$\cos(x(t))\lim_{\ell\to0} \frac{\sin(\ell h(t))}{\ell}.$$
What should I do next?