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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?2012-07-01
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    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.2012-07-01

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