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Thread: precompactness in L p and C[0,1]

  1. #1
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    precompactness in L p and C[0,1]

    Hi all,
    I'm trying to learn precompactness arguments , I have a question about precompactness in function spaces $\displaystyle L_{p} $ and C[0,1] :
    Prove the set $\displaystyle A = (sin(nt) )_{n=0}^{\infty} $ is precompact in $\displaystyle L_{p}(0,1)$ for $\displaystyle 1 \leq p < \infty $ , but it's not precompact in C[0,1]. I looked at the theorems about equicontinuity and similar stuff but I couldn't find anything useful , can anyone give a hint??
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  2. #2
    Super Member girdav's Avatar
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    Hi,
    if I denote by $\displaystyle f_n$ the function $\displaystyle t\mapsto \sin (nt)$ we see that $\displaystyle f_n\left(\frac{\pi}n\right) = \sin \pi =0$ and $\displaystyle f_n\left(\frac{\pi}{2n}\right) = \sin \frac{\pi}2 =1$ so the $\displaystyle \delta$ in the definition of uniform continuity has to be smaller than $\displaystyle \frac{\pi}{2n}$. Hence we can not have the same $\displaystyle \delta$ for the whole family. $\displaystyle A$ is not equi-continuous so $\displaystyle A$ is not precompact.
    To show the precompactness in $\displaystyle L^p$, I think you can use a criterion of equi-continuity in the mean for $\displaystyle L^p$.
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