Generally accepted notation for referencing function without defining it. Let $F\subseteq (\mathbb R \to\mathbb R)$ be some space of functions, and let $G:F\to \mathbb R$ be a functional. I have a statement of the following form:
$$\begin{align}\text{Let } &f^*(x):=x^2. \quad\quad\quad
 \text{Then }\\
&f^*\in \arg\max_{f\in F} G(f)
\end{align}$$
Rather than first defining a function and then referencing it, I'd like to compress this into one equation for brevity's sake. Something like:
$$(x\mapsto x^2)\in \arg\max_{f\in F} G(f)$$
Is there a generally accepted notation like this? I'd prefer not to invent something new and unknown.

I don't understand either of your statements, so both of them are too concise to be readable. 
Do you mean that $f^*$, which is $\underset{f \in F}{\operatorname{argmax}}  G(f)$, turns out to be the function defined by $f^*(x) = x^2$? If so, for the sake of comprehensibility rather than brevity, you should write this out in words in a complete sentence: for example,

Let $f^* = \underset{f \in F}{\operatorname{argmax}}  G(f)$. Then it turns out for mysterious reasons that $f^*(x) = x^2$.

(Possibly with an explanation why this is the function that maximizes $G(f)$.)
Or possibly (after the recent edits, this seems closer to the sort of emphasis you want):

Let $f^* \in F$ be given by $f^*(x) = x^2$. Then $f^* \in \underset{f \in F}{\operatorname{argmax}}  G(f)$.

