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$\Omega$ is a bounded domain of $\mathbb R^n$. If $\{f_n\}\subset L^p(\Omega)$ and $f_n\rightarrow f\in L^p(\Omega)$ weakly, then
$$\|f\|_p\leq \lim_{n\rightarrow\infty}\inf\|f_n\|$$

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    $\begingroup$ What exactly do you mean by $f_n \to f \in L^p(\Omega)$? Pointwise convergence a.e.? $\endgroup$ – Martin May 20 '13 at 10:29
  • $\begingroup$ Sorry I missed some information. It should be weakly convergence. $\endgroup$ – Falang May 21 '13 at 7:44
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You are asking to prove why $$\left(\int |f|^p\right)^{1/p} \leq \lim_{n\rightarrow \infty} \inf \left(\int {|f_n|^p}\right)^{1/p},$$

so it is enough to prove that $\left(\int |f|^p\right)\leq \lim_{n\rightarrow \infty} \inf \left(\int {|f_n|^p}\right)$. Now if $f_n \to f$ pointwise a.e. then $|f_n|^p \to |f|^p$ pointwise a.e. Thus $$\begin{eqnarray*} \int |f|^p &=& \int \lim_{n \rightarrow \infty} |f_n|^p \\ &=&\int \lim_{n \rightarrow \infty} \inf |f_n|^p \\ &\leq& \lim_{n \rightarrow \infty} \inf \int |f_n|^p \end{eqnarray*}$$

where the last step was made using Fatou's lemma. Q.E.D.

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  • $\begingroup$ Sorry I missed some information. It should be weakly convergence. $f_n\rightarrow f$ weakly in $L^p$. $\endgroup$ – Falang May 21 '13 at 7:45

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