EDB β€” 10J

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Exercises

  1. [10J] Having fixed \(s,t∈[1,∞]\) with \(s{\lt}t\), show that \(n^{-1/s}\| x\| _ s≀ n^{-1/t} \| x\| _ t\) (where we agree that \(n^{-1/∞}=1\)). (Note that this is an inequality between averages).
    (Hint. Set \(𝛼=t/s\) and \(y_ i=|x_ i|^ s\), then use the convexity of \(f(y)=y^{𝛼}\). Another tip: use [10M].)

    Solution 1

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Bibliography
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  • normed vector space
  • \( \Vert \cdot \Vert _p\) , in \( ℝ ^n\)
  • \( \Vert \cdot \Vert _\infty \) , in \( ℝ ^n\)
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