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Show the following define norms on R2

  1. Feb 24, 2010 #1
    norm (x) = abs(x1) + abs(x2)

    norm (x) = 2abs(x1) + 3abs(x2)

    It satisfied the first two properties, but I'm having trouble showing the Triangle Inequality is true. Proving the Triangle Inequality for the Euclidean norm is easy because you can get both sides into the Cauchy-Schwartz Inequality. However, I can't get these in that form. I'm wondering, though, if I could use the absolute value sum inequality to simply show it's true since the vectors are added component-wise.

    abs(A + B) < /equal to abs(A) + abs(B)
     
  2. jcsd
  3. Feb 24, 2010 #2

    radou

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    Let x = (x1, x2) and y = (y1, y2). Then, by definition ||x+y|| = |x1 + y1| + |x2 + y2|. Now simply use the triangle inequality for the standard Euclidean norm.
     
  4. Feb 24, 2010 #3
    I'm supposed to show the triangle inequality is true for this definition of a norm.
     
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