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

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?

"$\min$" 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.

• Okay thanks, but how does it make sense to state "$f(u) = \text{min}!$", what are when taking the argmin of? – user116403 Jan 21 '15 at 16:50
• It looks like sloppy writing. Then again, I don't have context of this. You have those notes. Is the exclamation point used anywhere else? What else is written there? Do you have a link to the notes. – user147263 Jan 21 '15 at 17:57
• Yes the exclamation is used again. From Zeidler's book "Nonlinear Functional Analysis and its Applications II/B" page 506. Do you know this book? – user116403 Jan 21 '15 at 18:03
• It looks like it should be $\text{min}f(u)$. – user116403 Jan 21 '15 at 19:42
• I heard of the book, but I don't have it lying nearby. If you post a screenshot of the page with that fragment, that might help. – user147263 Jan 21 '15 at 19:51