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Thread: Sequence, Lebesgue Space

  1. #1
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    Sequence, Lebesgue Space

    Prove that if $\displaystyle (g_n)$ is a sequence in $\displaystyle L^2[0, 1]$ with $\displaystyle || g_n ||_2 \leq 1$ for all $\displaystyle n \geq 1$ then $\displaystyle (g_n/n)$ converges to zero a.e.


    This should follow from this:

    If $\displaystyle (f_n)_n$ is a sequence in $\displaystyle L^1[0, 1]$ is such that $\displaystyle \sum_{n=1}^{\infty} || f_n ||_1 < \infty$ then $\displaystyle \sum_{n=1}^{\infty} | f_n(s) | < \infty$ for almost every $\displaystyle s \in [0, 1]$.


    However, I do not see how this would follow. How do I prove this?
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  2. #2
    MHF Contributor

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    Why not apply the quoted result to $\displaystyle f_n=\frac{g_n^2}{n^2}$?... Try to find how to conclude from there.
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