What does $f(u)=\min!$ mean in calculus of variations?

Question

I have a very simple notation related question. There are notes to calculus of variations [specifically: Zeidler's book "Nonlinear Functional Analysis and its Applications II/B" page 506] which states that we can consider the equation $$f'(u)= 0~~~~~ \text{ for } u \in X,$$ together with the corresponding minimum problem $$f(u) = \min!~~~~ \text{ for } u \in X.$$

Has anyone encountered the notation "min!"? What does it mean exactly and is there an alternative notation?

Answer 1

"$\min$" with an exclamation point could be used for various things:

• argmin, the point where minimum is attained
• as an indication that the minimum is exists
• for something else, e.g., $\min!$ could be defined to be the number $2\pi$

After browsing the book in question (Zeidler, "Nonlinear Functional Analysis and its Applications II/B"), I found that the author uses $\min!$ only in the context

Consider the variational problem $f(u)=\min!$ where $u$ is in some space, and some constraints hold.

So, for him it is just a shortcut for writing "minimize $f$".

I also remember some people writing "$f(u)\to \min$" to express the same thing.

It would be much better to spell it out in words.