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Let $(f_n)$ be functions in $L^p(\Omega), 1<p<\infty$ such that $(f_n) \rightarrow f$ almost everywhere and $\Vert f_n\Vert_p \rightarrow \Vert f\Vert_p$. How does one show that $f_n \rightarrow f$ weakly in $L^p(\Omega)$ without first showing that $f_n \rightarrow f$ strongly in $L^p(\Omega)$? (I know that under these hypotheses we in fact get that $f_n \rightarrow f$ strongly, as explained here

If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$ ,

but there must be an easier, direct argument that $f_n \rightarrow f$ weakly.)

I know that some subsequence $(f_{n_k})$ converges weakly to some $g$ in $L^p(\Omega)$ since the $f_n's$ are bounded, but then (1) how do we pass from the subsequence to the original sequence, and (2) how do we show that $g=f$?

Thanks!

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  • $\begingroup$ when I said "the $f_n$'s are bounded," I of course meant that the $\Vert f_n \Vert$'s are bounded. $\endgroup$ – jake Nov 2 '11 at 23:44
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    $\begingroup$ The proof of strong convergence is about three lines, and your proposed argument invokes Alaoglu's theorem, which in turn invokes Tychonoff's. I'm not sure I would call that "easier". $\endgroup$ – Nate Eldredge Nov 3 '11 at 0:10
  • $\begingroup$ @jake: You can edit your question using the "edit" link just above the comments, you know. $\endgroup$ – Henning Makholm Nov 3 '11 at 0:11
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The boundedness of $\|f_n\|_p$, together with the reflexiveness of $L^p(\Omega)$ for $p\in (1,\infty)$, allows us to conclude that $f_{n_k}\rightharpoonup g$ weakly in $L^p(\Omega)$ for some $g\in L^p(\Omega)$. Recall that $f_n\rightarrow f$ a.e. in $\Omega$. Using these facts and Egorov's theorem we can show $g=f$ a.e. in $\Omega$. Then via arguing by contradiction (a standard way) one can show the original sequence $f_n\rightharpoonup f$ weakly in $L^p(\Omega)$.

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Because the Norms of the f_n are bounded. There is a subsequence that converges weakly to some g in Lp: by Alaoglu's theorem the Normed ball is compact in the weak star topology. I think you can repeat this for any subsequence, and using the fact that fn conv ae to f, get that fn conv weakly to fn.

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