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Space l_infty(R) with sup norm

  1. Apr 7, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that the space l_\infty (R) of bounded sequences with the sup norm (ie x=(x_n) in l_\infty (R) )is not a inner product space.


    3. The attempt at a solution

    Using definition of parallelogram ||x+y||^2+||x-y||^2=2(||x||^2+||y||^2) (1)

    Consider x_n=1^-n and y_n=4-1/n then

    ||x||_\infty=sup|x_n| = 1 where n=1 and |y||_\infty=sup|y_n|=4 where n=\infty

    ||x+y||_\infty=|1+4|=5 where I chose n=1 for x_n and n=\infty for y_n so that we get the maximum value.

    ||x-y||_\infty = |0-4|=4 where I chose n=\infty for x_n and n=1 for y_n to get a maximum

    ||x+y||^2_\infty+||x-y||^2_\infty=41

    THis is not equal to the RHS 2(||x||^2+||y||^2)= 34

    implies the space l_\infty(R) is not an inner product space..?
     
  2. jcsd
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