1

What points should I prove when I am asked to prove that a particular norm, say $||\mathbf x||=||(x,y)||=(|x|^{1/2}+|y|^{1/2})^2$, is a norm in $\mathbb R^2$?

P.S I have read about the difference between a metric and a norm (Please refer to the answers to this question).

Here I have deliberately used the term norm instead of metrics; is it a mistake in this particular case? I have written the question this way because it is exactly what I read in a past math exam that I'm studying.

Thank you.

Community
  • 1
Charlie
  • 1,492
  • 2
    you should check the definition of norm. See http://en.wikipedia.org/wiki/Norm_%28mathematics%29. – Bobby Jan 27 '14 at 07:57
  • @Babgen Ok, so I need to verify absolute homogeneity, positivity, triangle inequality and the zero vector. What about my use of the term norm in the question? Is it correct? Should I have used the term metric? – Charlie Jan 27 '14 at 08:01
  • Yes, It is fine. – Bobby Jan 27 '14 at 08:15

1 Answers1

1

Check the following statement:

$1$.Absolute homogeneity: $||ax||=a||x||$, for $a\in \mathbb R$ and $x\in \mathbb R^2$.

$2$.Triangle inequality: $||x+y||\leq ||x||+||y||$.

$3$.Separates points: If $||x||=0$ then $x$ is the zero vector.

For Triangle inequality: $||(2,2)+(1,3)||=16$ but $||(2,2)||+||(1,3)||=15.71$. This cant be a norm.

Bobby
  • 7,396
  • I was reviewing this answer. How should I prove that the triangle inequality does not hold here without plugging some numbers in $||\mathbf x + \mathbf y||$? Should I use algebra to find a contradiction? – Charlie Aug 14 '14 at 10:16