1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Proof of a normed space

  1. Sep 17, 2008 #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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted