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!

Inner (dot) products

  1. Mar 19, 2008 #1
    1. The problem statement, all variables and given/known data

    For [tex]x,y \in R^n[/tex], their inner ("dot") product is given by

    [tex]<x,y>=\sum_{i=1}^n x_i y_i.[/tex]

    Also, we write


    2. Relevant equations

    Fix p>1. Show that for all [tex]x,y \in R^n[/tex] we have

    [tex]< \|x\|^{p-2}x -\|y\|^{p-2}y, x-y> \geq 0[/tex]

    3. The attempt at a solution

    Expanding the left-hand side, we can write

    [tex]<\|x\|^{p-2}x,x> -<\|y\|^{p-2}y,x>-<\|x\|^{p-2}x,y>+<\|y\|^{p-2}y,y>[/tex]

    which further simplifies to

    [tex]\|x\|^p +\|y\|^p -(\|x\|^{p-2} +\|y\|^{p-2})<x,y>.[/tex]

    Then I'm stuck. How do I show that the above is nonnegative?
    Last edited: Mar 19, 2008
  2. jcsd
  3. Mar 19, 2008 #2


    User Avatar
    Gold Member

    how about using cauchy-schwartz inequality?
    something like this
    then you get that what you wrote is greater than:
    now if ||x||>=||y|| then ||x||^p-1>=||y||^p-1
    I leave you to check this proposition.
  4. Mar 19, 2008 #3
    I've got it now, many many thanks!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook