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Math Help - 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  L_{p}  and C[0,1] :
    Prove the set A = (sin(nt) )_{n=0}^{\infty}  is precompact in  L_{p}(0,1) for 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 f_n the function t\mapsto \sin (nt) we see that f_n\left(\frac{\pi}n\right)  = \sin \pi =0 and f_n\left(\frac{\pi}{2n}\right) = \sin \frac{\pi}2 =1 so the \delta in the definition of uniform continuity has to be smaller than \frac{\pi}{2n}. Hence we can not have the same \delta for the whole family. A is not equi-continuous so A is not precompact.
    To show the precompactness in L^p, I think you can use a criterion of equi-continuity in the mean for L^p.
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