Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What does notation

$$\inf\int f(x) \,\mathrm{d}x$$

stand for? I noticed it in a question on this site.

Any keywords, links or stories about this or similar notations will be appreciated :)

Sorry for such a basic question, but i can't find a reasonable keyword to look this up anywhere.

share|improve this question
    
The infimum en.wikipedia.org/wiki/Infimum –  Jeremy Jan 20 '13 at 1:43
2  
The notation, exactly as put, does not make sense. I hope that it was not used. –  André Nicolas Jan 20 '13 at 1:44
    
@André - I have linked the usage that I saw. If that is correct and my usage is wrong, please specify. I just wanted to write some more simple/general example instead of that double integral. –  Juris Jan 20 '13 at 1:51
    
@AndréNicolas In the original question, the infimum was taken over all functions in a given space. In that context, the usage made sense. –  Ayman Hourieh Jan 20 '13 at 1:51
add comment

3 Answers

up vote 1 down vote accepted

It means infimum, which is a generalization of the notion of minimum. For example, $f(x)=1/x$ doesn't have a minimum on $x\in[1,\infty)$, but its infimum is $0$.

In the context in which it was used,

$$\inf \iint\limits_{x^2+y^2\leqslant1}\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2\mathrm dx\,\mathrm dy$$ for $C^\infty$ functions $u$ that... [more conditions on $u$]

it means the tightest bound on how small the value of the integral can get for any such function $u$.

share|improve this answer
add comment

To write $\displaystyle\inf\int f(x)\,dx$ without the words that came after that fails to convey what was said. It says "for $C^\infty$ functions that vanish at $0$ and [. . . . .]", and the expression inside the integral has "$u$" in it. In other words, it identifies a particular set. An infimum is an infimum of a set.

The infimum of a set with a smallest member is the smallest member. Thus the infimum of the set of all nonnegative numbers is $0$. The infimum of some sets with no smallest member exists. For example, there is no smallest positive number, and the infimum of the set of all positive numbers is $0$.

The infimum of a set is the greatest lower bound of the set. $0$ is a lower bound of the set of all positive numbers because $0$ is less than or equal to every positive number. No number bigger than $0$ is a lower bound of the set of all positive numbers. So $0$ is the greatest among all lower bounds.

share|improve this answer
add comment

$$\inf \int f(x)dx$$ is the largest real number that is less than or equal to $$\int f(x) dx$$ inf stands for infimum. See more information here:

http://en.wikipedia.org/wiki/Infimum

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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