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I think you have something like the following inequality in most finite dimensional spaces or sufficiently restricted infinite dimensional space:

$$ \|g\|_{\infty}\lt C\|g\|_2$$

where $C$ would probably depend on the dimension of the norm space or whatever restrictions you might have on the infinite dimensional space to allow this kind of inequality.

What is this called and what are some references and any simple online depictions that I can point people to? I may move this to math.stackexchange if that is possible.

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migrated from mathoverflow.net Jun 25 '13 at 16:26

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For a finite dimensional space, this holds for any two norms. Just compare the unit balls. –  Deane Jun 20 '13 at 18:15
    
Yes, "unit balls" is probably as good a reference as any. Hopefully, I can find a nice reference to close this question with. This is probably covered in some introductory chapter of an approximation theory book ... –  mathtick Jun 23 '13 at 14:48
    
possible duplicate of When $L\_p = L\_q$? –  Nate Eldredge Jun 25 '13 at 16:27

2 Answers 2

up vote 3 down vote accepted

As I discussed in this answer, this inequality holds precisely when the measure space is atomic, i.e. $$\inf\{\mu(A) : \mu(A) > 0\} > 0.$$

I don't think it has a name, per se. One way to prove it is just to observe that, as in Markov's inequality, we have for any $a$, $$\int f^2 \ge a^2 \mu(\{f \ge a\}).$$ For any $a < \|f\|_\infty$, we have $\mu(\{|f| \ge a\}) > 0$, hence $\mu(\{|f| \ge a\}) \ge m$. So for such $a$, $\int f^2 \ge a^2 m$. Now letting $a \uparrow \|f\|_\infty$ we get the desired result with $C = m^{-1/2}$.

If you like, you could write something like "by Markov's inequality" but that might just confuse the issue. I would probably write something like "because the measure is atomic, $\|f\|_\infty \le C \|f\|_2$" and leave it at that; the fact is very familiar for $\ell^p$ spaces (where $\mu$ is counting measure, so $m=1$) and the reader should be able to see that it holds for other atomic spaces. You can give the quick proof if you really feel it's necessary, and assign it an equation number if you need to refer to it often. It's a basic enough fact that I don't think a reference (or "simple online depiction", whatever that may be) is necessary.

In the finite dimensional case, as in Deane's comment, all norms are equivalent, and so I would just write "by equivalence of norms, $\|f\|_\infty \le C \|f\|_2$".

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I think it could be called as "local boundedness" in De Giorgi -Nash- Moser theory in PDEs.

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