Let $E$ be a real Banach space included in the space of functions $C^1$ of $\mathbb R\to \mathbb R $ and and $T:E\to E$ defined by $T(f)=f\circ f$. I am looking for an example of space E such that T is differentiable on E.
By developing, $T(f+h)$ we find
$$T(f(x)+h(x))=f(f(x))+h(f(x))+f'(f(x)).h(x)+h'(f(x)).h(x)+o(h(x))$$ but the term $h'(f(x)).h(x)$ is not linear on h and it seems that $DT_{f}\left(h\right)=\left(f'\circ f\right)\times h+\left(h\circ f\right)$ but we need a space where $(h'\circ f).h=o(h)$
I thought of the subspace of $C^{\infty}(\mathbb R, \mathbb R)$ formed of bounded functions of which all derivatives are bounded, provided with the norm $f\mapsto \|f\|_{\infty} + \|f'\|_{\infty}$ but this space is not complete space
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As suggested by Giorgio Metafune , I start with Gateaux differentiability . Let $E=C_b$ (bounded and continuous functions with the sup-norm). Let $g\in E$ and f be a function derivable on $\mathbb R$. By developping $$ (T(f+tg)-T(f))(x)=f(f(x)+tg(x))+tg(f(x)+tg(x))-f(f(x)) $$ then $$\lim_{t\to0}\frac{(T(f+tg)-T(f))(x)}{t}=\lim_{t\to0}\frac{f(f(x)+tg(x))-f(f(x)) }{t}+g(f(x))\\= f'(f(x)).g(x)+g(f(x)) $$
Then $$\qquad D_G(T)f(g)=g (f'\circ f)+g\circ f$$ and $g\to D_G(T)f(g)$ is linear. It's must be continuous to deduce Frechet differentiability
Update question Giorgio Metafune suggests considering $E=C_0(\mathbb R)$
Let $E=C_0(\mathbb R)$ and $T: E\to E$ defined by $T(f)=f\circ f$. Show ( if true) that T is Fréchet differentiable at all $f\in C^1(\mathbb R)$. Can some one help me or give answer
