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If $f, g$ are two functions on a bounded subset of $\mathbb R$, is there a bound on $\|f-g\|_2$, involving only $\|f-g\|_1$, $\|g\|_2$, and some other finite quantities? Here, $\|\cdot\|_p$ is the $L^p$-norm.

Thanks!

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    A bound that depends on what?2012-10-22
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    I am not sure I understand your question. Basically, I want to show that if $\|f-g\|_1$ is finite and $\|g\|_2$ is finite, then $\|f-g\|_2$ is finite.2012-10-22
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    That is not true in general: $f(x)=\sqrt{x}$, $g=0$.2012-10-22

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