Norm on ℝ² with ||(0,1)||=1=||(1,0)|| and ||(1,1)||=0.000001

[cummings12332]

Homework Statement

find a norm on R2 for which||(0,1)||=1=||(1,0)|| but ||(1,1)||=0.000001

Homework Equations

hints: ||(a,b)|| = A |a+b|+B|a-b

The Attempt at a Solution

by the hints i have A+B=1 and 2A=0.000001
then solved the equations system i get A=0.0000005 B=1-A=0.9999995 then ||(a,b)|| = 0.000001|a+b|+0.9999995 |a-b|

is it the answers?

 

[Mark44]

Homework Statement

find a norm on R2 for which||(0,1)||=1=||(1,0)|| but ||(1,1)||=0.000001

Homework Equations

hints: ||(a,b)|| = A |a+b|+B|a-b

The Attempt at a Solution

by the hints i have A+B=1 and 2A=0.000001
then solved the equations system i get A=0.0000005 B=1-A=0.9999995 then ||(a,b)|| = 0.000001|a+b|+0.9999995 |a-b|

is it the answers?

Yes, these values satisfy the given conditions.

 

[jbunniii]

Another solution would be to use a p-norm:
$$||(a,b)|| = (a^p + b^p)^{1/p}$$
with $p \geq 1$. This will satisfy $||(1,0)|| = ||(0,1)|| = 1$ for any $p$, so all you have to do is solve for the $p$ which gives the desired result for $||(1,1)||$.