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If both the $L^{2}(\mathbb R)$ norm and $L^{\infty}(\mathbb R)$ norm of a function $f$ are finite, is there any relation between the two norms in this case?

I know that there is a relation in case of a set of finite measure $S$, (i.e., $L^{2}(S)$ norm and $L^{\infty}(S)$), but what about the $\mathbb R$ case? If in general there is no relation, when we could have it?

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Since your comment indicates that David's answer doesn't settle it for you, could you please make your question more precise? What do you mean by relation? –  Jonas Meyer Jul 1 '12 at 4:12
    
Finite measure gives you $\|f\|_2\leq C\|f\|_\infty$ (for some constant $C$ independent of $f$). If there is a positive lower bound on the measures of sets of positive measure, then $\|f\|_\infty\leq C\|f\|_2$ (for some constant $C$ independent of $f$). For $\mathbb R$ you have neither of these, as David's answer shows. –  Jonas Meyer Jul 1 '12 at 4:17

1 Answer 1

No. Consider characteristic functions on sets of the form $[-n,n,]$. These have infinity norm of one but large $L_2$ norm. On the other hand, consider functions of the form $f=n^{1/4}\chi_{[0,1/n]}$. These have small $L_2$ norm but large infinity norm.

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Okay, I didn't think of examples of sequences, but if I allow sequences then I should assume that the limit of $\infty$ norms and the limit of the $L^2$ norms are both exists and finite. So, do we still have no relation? –  Seamon Jul 1 '12 at 3:06

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