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Show the norm ||x|| is less or equal to A|x| for some constant A

  1. Oct 2, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that [tex]\|x\| \leq A|x| \forall x \in \mathbb{R},[/tex] where [tex]A \geq 0.[/tex]

    2. Relevant equations
    We know the norm is a function [tex]f: {\mathbb{R}}^{d} \to \mathbb{R},[/tex] such that:
    [tex]a) f(x) = 0 \iff x = 0,[/tex]
    [tex]b) f(x+y) \leq f(x) + f(y),[/tex] and
    [tex]c) f(cx) = |c|f(x) \forall c \in \mathbb{R}[/tex]

    3. The attempt at a solution
    Ugh, I'm completely stumped here, and don't know where to begin. I know that for the Euclidean norm this is trivial, but I don't know how to even begin showing this in general. In particular, I don't see where I could grasp the absolute value (or the dot product), or how to start comparing it to the norm.

    Any help would be greatly appreciated.
     
  2. jcsd
  3. Oct 3, 2011 #2
    I'll just post a hint : try A=|f(1)|.
     
  4. Oct 3, 2011 #3
    Hey, thanks for the hint, although I actually figured it out yesterday a couple of hours after posting :smile: I did it via the unit vectors and then expanding upon the properties b) and c). I assume this is what you were going for with the hint, as well, right?
     
  5. Oct 4, 2011 #4
    Yes, I thought on similar lines.
     
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