Is it acceptable to write $\sup\,\{f(x) \mid x \in \mathbb R\}$ (where $f\colon \mathbb R \to \mathbb R$) as $\sup\,\{f(x)\}_{x \in \mathbb R}$? I have seen it written as $\sup\limits_{x \in \mathbb R}f(x)$, but never as $\sup\,\{f(x)\}_{x \in \mathbb R}$. It seems as though it is acceptable, but having never seen it, I am unsure. Sorry if this is a bad question.
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That is acceptable, though it looks kind of ugly to my eyes. I would say that $$\sup\limits_{x \in \mathbb R}f(x)$$ is preferable for stylistic reasons, but $\sup\,\{f(x)\}_{x \in \mathbb R}$ is not mathematically incorrect.
Jim
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I wouldn't, not because I think it's impossible to justify, but because no one writes it that way. I personally prefer $\sup\{f(x)|x\in\mathbb R\}$ (or just $\sup\{f(x)\}$!) to $\sup_{x\in\mathbb R}\{f(x)\}$, because $\sup$ is a function giving you the least upper bound of a set, and shouldn't have to be parameterized by anything. But both those two usages are common, and will be understood immediately, whereas $\sup\{f(x)\}_{x\in\mathbb R}$. In the end, it's not going to take up an enormous amount of your time to write $\sup\{f(x)|x\in\mathbb R\}$ each time: a lot of mathematical clarity can be lost by pointless abbreviations.
Incidentally, you can also use a colon $:$ rather than a pipe $|$ ($\sup\{f(x):x\in\mathbb R\}$), though I think that that is entirely a matter of taste.
John Gowers
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