Suppose we have $f \in L^2(\Omega)$ and $v \in D(\Omega)$ a test-function. If we multiply $f$ by $v$, the $(fv)$ function belongs to which space?
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This not a meaningful question. There are all kinds of spaces to which the product belongs. – Kavi Rama Murthy Apr 29 '20 at 23:23
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i mean the product fv belongs to $L^2(\Omega)$, or $L^1(\Omega)$ ? – kayzo Apr 29 '20 at 23:39
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thank you for helping me ! – kayzo Apr 29 '20 at 23:39
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$fv \in L^{2}$ because $v$ is bounded; $fv \in L^{1}$ by Holder's inequality: $\int |fv| \leq \sqrt {\int |f|^{2} } \sqrt {\int |v|^{2} }$.
Kavi Rama Murthy
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@kayzo $\int|fv|^{2} \leq M\int |f|^{2}$ if $M$ is a bound for $|v|^{2}$. – Kavi Rama Murthy Apr 29 '20 at 23:50
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Downvoting by revenge. Someone downvoted several of my answers simultaneously. If the answer is clear to you, you can consider upvoting/ approving the answer. – Kavi Rama Murthy Apr 30 '20 at 05:43