I am currently studying Newton's method on Banach spaces, specifically as it pertains to the linearization of nonlinear PDEs. In a sample problem, the following operator between Banach spaces came up:
$$ \begin{align*} F: C^2_0[0,1] &\longrightarrow C^2[0,1] \\ u &\mapsto [ t \mapsto u''(t) - f(t,u(t)) ] \end{align*} $$
where $f: [0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function that is continuously differentiable in the second argument.
This said to have a Fréchet derivative $F'(u)$, for all $u \in C^2_0[0,1]$, which when taken as a linear operator, maps any other function $y \in C^2_0[0,1]$ to the function defined by
$$ [F'(u)](y)(t) = y''(t) - \frac{\partial f(t,u(t))}{\partial u} y(t)$$
I am trying to show that this is indeed true, but I have come up against a problem: the Fréchet derivative $F'(u)$ is supposed to be a bounded linear functional, yet it seems to me that it is possible to construct a sequence of functions in $C^2_0[0,1]$ which are bounded in the sup norm, but whose second derivative (and therefore the expression in the derivative above) grows beyond all limits in that same norm.
Specifically, the series of test functions
$$ \rho_{\epsilon}(x) := exp(-\frac{1}{1-(\frac{x-1/2}{\epsilon})^2})$$
for $\epsilon \rightarrow 0$ should do the trick, as $\| \rho_{\epsilon} \|_{C^2_0[0,1]} = 1/e$ for all $\epsilon$, yet $\rho''_{\epsilon}(1/2) = \frac{-2}{\epsilon^2e}$ grows arbitrarily large in the norm.
Can anyone see what I am missing? I don't think the book would make such a mistake.