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Find a norm on R2

  1. Nov 16, 2012 #1
    1. The problem statement, all variables and given/known data
    find a norm on R2 for which||(0,1)||=1=||(1,0)|| but ||(1,1)||=0.000001




    2. Relevant equations
    hints: ||(a,b)|| = A |a+b|+B|a-b


    3. The attempt at a solution
    by the hints i have A+B=1 and 2A=0.000001
    then solved the equations system i get A=0.0000005 B=1-A=0.9999995 then ||(a,b)|| = 0.000001|a+b|+0.9999995 |a-b|

    is it the answers???
     
  2. jcsd
  3. Nov 16, 2012 #2

    Mark44

    Staff: Mentor

    Yes, these values satisfy the given conditions.
     
  4. Nov 16, 2012 #3

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Another solution would be to use a p-norm:
    [tex]||(a,b)|| = (a^p + b^p)^{1/p}[/tex]
    with [itex]p \geq 1[/itex]. This will satisfy [itex]||(1,0)|| = ||(0,1)|| = 1[/itex] for any [itex]p[/itex], so all you have to do is solve for the [itex]p[/itex] which gives the desired result for [itex]||(1,1)||[/itex].
     
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