The discussion focuses on finding a norm on R² such that ||(0,1)||=1=||(1,0)|| and ||(1,1)||=0.000001. The solution involves defining the norm as ||(a,b)|| = A |a+b| + B |a-b|, leading to the equations A+B=1 and 2A=0.000001. The values A=0.0000005 and B=0.9999995 satisfy the conditions. Additionally, a p-norm can be utilized, where ||(a,b)|| = (a^p + b^p)^(1/p) for p ≥ 1, allowing for further exploration of p to meet the ||(1,1)|| requirement.
PREREQUISITESMathematicians, students studying linear algebra, and anyone interested in functional analysis and vector space norms.
Yes, these values satisfy the given conditions.cummings12332 said: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?