- #1
Ryker
- 1,086
- 2
Homework Statement
Show that [tex]\|x\| \leq A|x| \forall x \in \mathbb{R},[/tex] where [tex]A \geq 0.[/tex]
Homework 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]
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.