Generally accepted notation for referencing function without defining it.

Question

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.

Answer 1

If you don't want to invent something new you could write $$ \big(\arg\max_{f\in F} G(f) \big)(x) = f^\star(x) = x^2. $$ However, in my opinion it would be preferable to write it as $$ f^*=\arg\max_{f\in F} G(f),\; f^*(x)=x^2, $$ which has approximately the same length.

Answer 2

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 $f^* \in \underset{f \in F}{\operatorname{argmax}} G(f)$.