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Suppose $f(x)$,$xf(x)$ $\in$ $L_2(\mathbb{R})$. Prove that $f(x)\in$ $L_1(\mathbb{R})$.

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What have you tried? –  Clayton Jan 4 '13 at 15:52
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If this is homework, please tag it as homework. –  user1551 Jan 4 '13 at 15:53
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up vote 5 down vote accepted

Let $g(x)=\frac{1}{1+|x|}$ and $h(x)=(1+|x|)f(x)$. Clearly $g\in L^2(\mathbb{R})$. Your condition implies that $h\in L^2(\mathbb{R})$. Noting that $f=gh$, the conclusion follows from Cauchy-Schwarz inequality.

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