Is $C(C(\mathbb R))$ notation for the set of continuous functions mapping $C(\mathbb R)$ to itself?

Question

Given that in general functional analysis we have $C(\mathbb{R})$ being the set of all continuous functions, $f: \mathbb{R} \to \mathbb{R}$. However, could I use $C(C(\mathbb{R}))$ notationally to be the the set of all continuous functions, $f: C(\mathbb{R})\to C(\mathbb{R})$?

Answer 1

No, that is not standard and is not likely to be understood as intended. As Sam points out in a comment,

$C(M)$ is usually understood as $C(M,\mathbb R)$. So for your purpose a better notation should be $C(C(\mathbb R),C(\mathbb R))$.

Furthermore, echoing PVAL's comment, it is not obvious what topology you have in mind on $C(\mathbb R)$, and this needs to be specified in order to determine what is meant by "continuous" for functions to or from this space.