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I want to ask questions : First : In general how can I study the convergence of any sequence in $L^p$ ( $L^1$ , $L^2$ , $L^3$ , .......) of course the sequence belongs to the space which I want to study the convergence of the sequence in it . How can I determine if the sequence is convergent or divergent ??? Because I don't know what is the general method to study the convergence in like space

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  • $\begingroup$ Although this is neither necessary nor sufficient, it is probably useful to check for pointwise convergence first. Precise statement: If the sequence converges in $L^p$, then there is some subsequence which converges almost everywhere to the limit. Once the limit is identified, try to look for applicable convergence theorems (monotone convergence, dominated convergence, or the Vitali convergence theorem (see en.wikipedia.org/wiki/Vitali_convergence_theorem, although I don't really like the version on Wikipedia). $\endgroup$ – PhoemueX Feb 27 '15 at 19:33
  • $\begingroup$ Here (math.stackexchange.com/questions/116820/…) is a version of Vitali's theorem that I like more (since it also applies to infinite measure spaces). $\endgroup$ – PhoemueX Feb 27 '15 at 19:39
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I would say that your first port of call is the fact that these spaces are complete and so we can appeal to the fact that absolute convergence implies convergence in Banach spaces.

Edit: You should unaccept this because while it is correct it is referring to convergence of series rather than sequences... then I can delete it.

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