Let $E=C([0,1]^m,\mathbb{R}^n)$ where $K=[0,1]^m$ where $E$ has the compact convergence topology. Recall that for a function $f:[0,1]^m\rightarrow [0,1]^m$ the associated composition operator $C_f$ is defined by
$$
\begin{aligned}
C_f:E &\rightarrow E\\
g &\mapsto g\circ f.
\end{aligned}
$$
We recall that $C_f$ is call cylic if there is some $g^f\in E$ such that $\operatorname{span}\{g^f\circ f^n\}_{n=0}^{\infty}$ is dense in $E$.
Lastly, let $L(E)$ denote the space of bounded linear operators on $E$ with the weak operator topology.
Is the span of the set of composition operators $C_f$ for which $f$ is cylic, dense in $L(E)$? If yes, what about if it instead has the strong operator topology?
