Show the following define norms on R2

[Shackleford]

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)

 

[radou]

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.

 

[Shackleford]

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.

I'm supposed to show the triangle inequality is true for this definition of a norm.