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Thread: convergence in Lp space

  1. #1
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    convergence in Lp space

    Let $\displaystyle (g_n)$ be a sequence that does not convergence in $\displaystyle L_p$ to a function $\displaystyle f\in L_p$ and let $\displaystyle (h_n)$ be a subsequence of $\displaystyle (g_n)$ that convergence in $\displaystyle L_p$ to $\displaystyle f$.
    Does any contradiction exists?
    I know if it is the case where convergence is in $\displaystyle \mathbb{R}$,it is trivially true.But,what about in $\displaystyle L_p$?
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  2. #2
    Super Member Rebesques's Avatar
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    Also (trivially) true, consider $\displaystyle \{g_1,f,g_2,f,\ldots\}$
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