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- Jun 20, 2014

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Let $1 < p < \infty$, and let $(f_n)$ be a sequence of real-valued functions in $\mathscr{L}^p(-\infty, \infty)$ which converges pointwise a.e. to zero. Show that if $\|f_n\|_p$ is uniformly bounded, then $(f_n)$ converges weakly to zero in $\mathscr{L}^p(-\infty,\infty)$.

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