Notation in functions spaces

Question

I'm wondering if there is a convention for the following. Let $g:\mathbb{R}^2\to \mathbb{R}$ be given and u in $C^1(\mathbb{R})$. I'm looking for a notation for $g(x,u(x))$ is in the space of continuously differentiable functions. Usually one wants to get rid of the $x$, i.e. something like $g(.,u)\in C^1(\mathbb{R})$ or $g(.,u(.))\in C^1(\mathbb{R})$. Is one notation prefered or is there a different notation? If $u$ would also depend on time and $u\in C^1(\mathbb{R}^2)$ this seems to get messy. If it is clear that $g$ does only depend on $x$ I would prefer $g(.,u)\in C^1(\mathbb{R}^2)$ over $g(.,u(.))\in C^1(\mathbb{R}^2)$, since the "dot" has two meanings in the later expression. Any remark is appreciated.

Answer 1

not too keen on "dots" as placeholders.

there is a natural injection, which we might call $\iota_2^2:u \rightarrow (id, u)$ of $C^1(\mathbb{R})$ to $C^1(\mathbb{R}^2)$ - here the superscript gives the dimension of the codomain space, and the subscript signifies which place the image function occupies in the list of arguments. if you drop the diacritics as understood in the context, and use the index notation for the image under a map, then you could speak of $g \circ u^{\iota}$

as stated, this doesn't deal with the second part of your question, but there is still a little mileage in the underlying idea.