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I saw some proves here relying on "uniform boundedness", but "uniform boundedness" is aplly to complete spaces("Baire thorem").

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    What are you asking? – 2017-02-04
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    Weak convergence implies boundedness in norm vector space?I wanted to mentiond that I saw some proves here but I don't think them correct because of completeness. – 2017-02-04
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    This might help? http://math.stackexchange.com/a/695469/27978 – 2017-02-04
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    No because that's again a proof that using "uniform boundedness" which is realy working in that case because the space Lp is complete. – 2017-02-04
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    Read the last line of the answer. The point is that the dual is always complete. – 2017-02-04
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    thank you very much for your help! – 2017-02-04
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    You are very welcome, good luck! – 2017-02-04

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