Suppose $f(x)$,$xf(x)$ $\in$ $L_2(\mathbb{R})$. Prove that $f(x)\in$ $L_1(\mathbb{R})$.
2026-05-04 17:01:02.1777914062
Prove that $f(x)$ is integrable on $\mathbb{R}$.
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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.