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Ineqaulity on \ell_p

  1. Oct 19, 2009 #1
    Could anyone prove or disprove the following inequality:
    [itex]||x||_{p}\leq||x||_{p'}[/itex] for all [itex]x\in\mathbb{R}^{n}[/itex] if [itex]p'>p\geq1[/itex]?

    By the way, this is not a homework problem.

    Any help on this will be highly appreciated.
     
  2. jcsd
  3. Oct 19, 2009 #2

    berkeman

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    Staff: Mentor

    Where is the problem from?
     
  4. Oct 19, 2009 #3
    This a problem that I was curious about, because we know that [itex]||x||_{m}\leq||x||_{1}[/itex] for any positive integer [itex]m[/itex], and then I wondered if it is true for any [itex]p\geq1[/itex].

    But it would be great if one can show the following:
    [itex]||x||_{p}\leq||x||_{1}[/itex] for [itex]p\geq1[/itex],
    so could anyone help me on this?
     
    Last edited: Oct 19, 2009
  5. Oct 19, 2009 #4
    [tex]\|x\|^p_p = \sum_i{|x_i|^p} \leq \left( \sum_i{(|x_i|)} \right)^p = \|x\|_1^p[/tex]

    Though I am very dizzy right now, it should be OK where I used [itex]a^2+b^2 \leq (a+b)^2, a,b > 0[/itex]
     
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