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Functions not satisfying parallelogram identity with supremum norm

  1. Mar 3, 2010 #1
    1. The problem statement, all variables and given/known data
    Find two functions [tex]f, g \in C[0,1][/tex] (i.e. continuous functions on [0,1]) which do not satisfy

    [tex]2 ||f||^2_{sup} + 2 ||g||^2_{sup} = ||f+g||^2_{sup} + ||f-g||^2_{sup}[/tex]

    (where [tex]|| \cdot ||_{sup}[/tex] is the supremum or infinity norm)

    2. Relevant equations
    Parallelogram identity: [tex]2||x||^2 + 2||y||^2 = ||x+y||^2 + ||x-y||^2[/tex] holds for any x,y


    3. The attempt at a solution
    Honestly no idea.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 3, 2010 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Just try some functions. It's really not hard to find an example that doesn't work.
     
  4. Mar 3, 2010 #3
    For posterity, two functions which fit nicely are
    f(x) = x
    g(x) = x-1


    (I had tried lots of functions but they worked; not very helpful response)
     
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