Proof of a normed space

  1. 1. The problem statement, all variables and given/known data
    The norm is defined by [tex]\left\|x\right\|_{p}=\left[\sum^{\infty}_{k=1}\left|\alpha^{i}\right|^{p}\right]^{1/p}[/tex]
    where [tex]x=(\alpha^{1},\alpha^{2},....,\alpha^{n})[/tex]

    Prove that this is a norm on [tex]V_{\infty}(F)[/tex]
    2. Relevant equations
    All conditions satisfied for a normed space.


    3. The attempt at a solution
    Well, I proved the first condition which is

    i)
    [tex]\left\|x\right\|_{p}=\left\{\left|\alpha^{1}\right|^{p}+\left|\alpha^{2}\right|^{p}+.....+\left|\alpha^{n}\right|^{p}+....\right\}^{1/p}[/tex]

    This must be positive definitive, therefore [tex]\left\|x\right\|_{p}>0[/tex]

    On the second condition I don't know whether taking the product of this norm with a [tex]\beta\in F[/tex] since the sum is infinite. I got stuck at this point.

    And also I presume, for the third condition, Minkowski inequality cannot be used anymore to prove it.
     
  2. jcsd
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