I am aware that the implicit and inverse function theorems can be generalized to infinite dimensional cases, but I am having difficulty in applying it to a specific calculation.
Let $C^1_{\mathbb{R}}[0,1]$ be the space of real-valued $C^1$ functions on the interval $[0,1]$. If we impose the following norm: \begin{equation} \lVert f \rVert := \lVert f \rVert_{sup}+\lVert f' \rVert_{sup} \end{equation}$$ \begin{equation} \lVert f \rVert := \lVert f \rVert_{\sup}+\lVert f' \rVert_{\sup} \end{equation} $$ for $f \in C^1_{\mathbb{R}}[0,1]$, it is clear that $C^1_{\mathbb{R}}[0,1]$ is a Banach space over $\mathbb{R}$.
Similarly
Similarly, $C_{\mathbb{R}}[0,1]$ is the Banach space of real-valued continuous functions on $[0,1]$ with the supremum norm.
Then, we can think of the operator $F:C_{\mathbb{R}}^1[0,1] \to C_{\mathbb{R}}[0,1]$ defined by \begin{equation} F(f):=sinh(f)+(f')^2 \end{equation}
How$$ \begin{equation} F(f):=\sinh(f)+(f')^2 \end{equation} $$ How do I show that (or is it indeed true that) $F$ is strongly $C^1$ and the derivative at each point is a linear isomorphism? More generally, how about the case $f \to G(f,f')$ for any smooth function $G : \mathbb{R}^2 \to \mathbb{R}$ whose Jacobian determinant never vanishes?
I appreciate any help.
