What does the notation inf{...} mean?

Question

I came across

$$\inf\{k : f \in C^k\}$$

What does $\inf\{\cdot\}$ mean? I have been looking, but haven't found anything.

It doesn't make a lot of sense in this particular context, since if $f$ is $\mathcal C^k$ for any one $k$, then it is also $\mathcal C^0$, so this infimum is either $0$ or (perhaps) $\infty$. — hmakholm left over Monica, Jul 21 '18 at 11:23

Davide Morgante · Answer 1 · 2023-10-13T10:34:16.467

21

Suppose that you have a non-empty set of numbers,as an example $A = \{1,10,\pi,55, 11.2, \sqrt{2}, {1\over 2}\}$ then the infinium of this set is the greatest lower bound of the set. In this simple case $$\inf\{A\}={1\over 2}$$

As an added bonus: what you gave us $$\inf\{k: f\in C^k\}$$ means that, given a function $f$ differentiable $n$ times, the infimum is the lower possible $k$ such that the $k$-th derivative of this function is continuous. But I think that $k$ will be $0$ every time..

edited Oct 13 '23 at 10:34

answered Jul 21 '18 at 11:22

Davide Morgante

3,578

1

"lower upper bound"?? – hmakholm left over Monica Jul 21 '18 at 11:24
Oops, my bad, I'm correcting it – Davide Morgante Jul 21 '18 at 11:24
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Maybe you meant not "infinium" but "infimum" and not "lower bound" but "greatest lower bound" – Юрій Ярош Jul 21 '18 at 11:35
Thank you all, I wrote the answer quickly! Now it's all corrected – Davide Morgante Jul 21 '18 at 11:37
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And by axiom, every non empty down bounded set of real numbers has infimum... – dmtri Jul 21 '18 at 12:17
This should be the accepted answer as it's clear, concrete, and rich. Still, it's difficult when applying the concept to a new scenario. For example, does inf{$x$: $F_X{(x)}$ $\ge$ $y$} mean the lowest value of $x$ such that $F_X{(x)}$ $\ge$ $y$, @DavideMorgante? – Nemo Jan 09 '20 at 03:02

score 8 · Accepted Answer · answered Jul 21 '18 at 11:18

8

It means infimum. So, $\inf\{\ldots\}$ is the infimum of the set $\{\ldots\}$ (assuming that it is a non-empty set of real numbers with a lower bound).

answered Jul 21 '18 at 11:18

José Carlos Santos

440,053

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To save a google search: https://en.m.wikipedia.org/wiki/Infimum_and_supremum – Crosby Jul 21 '18 at 11:19
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This answer is a bit like Q “what does etc. mean” being answered with “etc. means et cetera.” I.e. factually correct but not really what the OP was seeking. If the OP knew what infimum was they would know what inf meant. – nerak99 Sep 21 '22 at 21:02

score 6 · Answer 3 · answered Jul 21 '18 at 11:30

Lower bound of some set of numbers is number which smaller or equal to any number of the set.
Greatest lower bound of some set of numbers is a number which is a lower bound of the set and is bigger or equal to any other lower bound of the set. $\inf A$ means greatest lower bound of the set $A$. So e.g. $\inf A = 5$ means greatest lower bound of the set $A$ is $5$. Greatest lower bound is also called infimum.

What does the notation inf{...} mean?

3 Answers3