I am reading Audin and Damian's book *"Morse theory and Floer homology"*. In Proposition 8.1.4 which reveals the transversality property of moduli space of solutions of Floer equation, the authors define an operator
$$
\begin{aligned}
\begin{aligned}\Gamma:W^{1,p}(\mathbf{R}\times S^1;\mathbf{R}^{2n})\times C_\varepsilon^\infty(H_0)\end{aligned}& \longrightarrow L^p(\mathbf{R}\times S^1;\mathbf{R}^{2n}) \\
(Y,h)& \longmapsto(dF^H)_u(Y)+\operatorname{grad}_uh.
\end{aligned}
$$
Here $C_\varepsilon^\infty(H_0)$ is the space of perturbations of Hamiltonian function $H $, F is the Floer map
$$\begin{aligned}&F:C^\infty(\mathbf{R}\times S^1;W)\longrightarrow C^\infty(\mathbf{R}\times S^1;TW)\\&u\longmapsto\frac{\partial u}{\partial s}+J\frac{\partial u}{\partial t}+\mathrm{grad}_u(H_t).\end{aligned} $$

My **question** is the exercise 44 of this book. Why is the kernel of $\Gamma$ is not finite-dimensional? I know that the difference between finite-dimensional space and infinite-dimensional space is that unit ball in infinite-dimensional space is not compact. But I don't know how to show it in the exercise. Any hint is welcome!

Hint.They show in Section 8.7c that $(dF^H)_u$ is a Fredholm operator. $\endgroup$