1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook