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Suppose $\text{sup}\|f_n\|_p <+ \infty$ and $f_n \to f$ almost everywhere on $E$. I want to prove weak convergence $f_n \rightharpoonup f.$

I already have a proof that there exists a sub-sequence $\{f_{n_i}\}$ such that $f_{n_i}\rightharpoonup f$ weakly in $L^p.$

So in can reformulate my question as:

How to show from the weak convergence of such sub-sequence the convergence of the whole sequence?

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You have to show that $\int f_ng \to \int fg$ for all $g \in L^{q}$ where $\frac 1 p+\frac 1 q=1$. For this it is enough to show that every subsequence of $(\int f_ng )$ has a subsequence converging to $\int fg$.

A sequence of real numbers $(a_n)$ converges to a real number $a$ iff every subsequence of $(a_n)$ has a further subsequence converging to $a$. [You can give a proof by contradiction].

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  • $\begingroup$ can you expand a little? I don't understand. $\endgroup$
    – user733151
    Dec 9 '19 at 12:25
  • $\begingroup$ @ann.boz I added some details when you were typing the comment. $\endgroup$ Dec 9 '19 at 12:26
  • $\begingroup$ and how does this prove what I want? $\endgroup$
    – user733151
    Dec 9 '19 at 12:26
  • $\begingroup$ I did not prove that every subsequence weakly converges, I just proved that there exists one that does so. So I don't see how you proved what I want to prove. $\endgroup$
    – user733151
    Dec 9 '19 at 12:29
  • $\begingroup$ I'm sorry I'm not understanding.. :( $\endgroup$
    – user733151
    Dec 9 '19 at 12:30

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