2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Now that I understand the question - there is no typical notation for this, because it is not common to do two things at once. If $f^*$ is defined to be the arg max, then there is a claim that $f^*$ is also $x^2$. If $f^*$ is defined to be $x^2$, there is a claim that it is the arg max. Either way, it is more common to separate the definition from the claim. More generally, the $\lambda x.x^2$ notation from computer science and the related $x \mapsto x^2$ notation are not especially common in math, with the possible exception of some specific subfields. $\endgroup$ – Carl Mummert Aug 31 '18 at 14:16
  • $\begingroup$ @CarlMummert, well, let's say that we DON'T want to be able to keep the function symbol $f^*$ around, then it's not really doing two things: essentially we would be saying in one single statement: "the function that maximizes $G(f)$ is the function that maps $x$ to $x^2$". $\endgroup$ – user56834 Aug 31 '18 at 14:18
  • 1
    $\begingroup$ !Programmer2134: so why not use that sentence? It is exceptionally clear. $\endgroup$ – Carl Mummert Aug 31 '18 at 14:19
  • $\begingroup$ @CarlMummert, because I want to be able to write it very often, with brevity $\endgroup$ – user56834 Aug 31 '18 at 14:20
  • $\begingroup$ I think there is no standard notation for that which would be immediately clear to everyone, since as I said the $\mapsto$ notation is not very common except possibly in specialized areas. $\endgroup$ – Carl Mummert Aug 31 '18 at 14:22
4
$\begingroup$

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

$\endgroup$
  • $\begingroup$ $\underset{f \in F}{\operatorname{argmax}} G(f)$ is not a set: it means "the element of $F$ which maximizes the value of the functional $G$", and elements of $F$ are functions. It equals the function that maps $x$ to $x^2$. $\endgroup$ – Misha Lavrov Aug 31 '18 at 14:12
  • $\begingroup$ !Misha - that is a reasonable interpretation here, but what if the maximum happens are multiple points? I was also looking at en.wikipedia.org/wiki/Arg_max . . But now I see - if the OP had written $f^* \in \text{arg max} G(f)$, where $f^*(x) = x^2$, it would be clearer to me. $\endgroup$ – Carl Mummert Aug 31 '18 at 14:13
  • 3
    $\begingroup$ @CarlMummert Quoth wikipedia: "If the maximum is reached at a single point then this point is often referred to as the arg max, and arg max is considered a point, not a set of points." This is the standard use case, in my opinion. (Though, of course, the fact that the maximum is reached at a single point - and that it is reached at all - must be justified.) $\endgroup$ – Misha Lavrov Aug 31 '18 at 14:14
  • 1
    $\begingroup$ +1 for use words. $\endgroup$ – Ethan Bolker Aug 31 '18 at 14:16
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.