- #1

Lambda96

- 189

- 65

- Homework Statement
- For which ##\lambda## does ##\Phi## have a norm in ##\mathbb{R^3}##

- Relevant Equations
- none

Hi,

The task is as follows

In order for it to be a norm, the three properties must be fulfilled.

1. Positive definiteness

2. Absolute homogeneity

3. Triangle inequality

##\textbf{Positive definiteness}##

Since all three elements are given in absolute value, the result of ##\max{}## will always be positive, no matter what value the ##\lambda## will have

##\textbf{Absolute homogeneity}##

##\max\{|s \cdot x|,|s \cdot y|^{\lambda}, |s \cdot z|\} = |s| \cdot \max\{|x|,|y|^{\lambda},|z|\}##

##\textbf{Triangle inequality}##

##|\max\{|x_1|,|y_1|^{\lambda},|z_1|\} +\max\{|x_2|,|y_2|^{\lambda},|z_2|\}| \leq |\max\{|x_1|,|y_1|^{\lambda},|z_1|\}| + |\max\{|x_2|,|y_2|^{\lambda},|z_2|\}|##

Now, I'm not sure, but ##\lambda## can now take on all values without the triangle inequality not being fulfilled. At least I can't think of an example where the triangle inequality is not fulfilled.

The task is as follows

In order for it to be a norm, the three properties must be fulfilled.

1. Positive definiteness

2. Absolute homogeneity

3. Triangle inequality

##\textbf{Positive definiteness}##

Since all three elements are given in absolute value, the result of ##\max{}## will always be positive, no matter what value the ##\lambda## will have

##\textbf{Absolute homogeneity}##

##\max\{|s \cdot x|,|s \cdot y|^{\lambda}, |s \cdot z|\} = |s| \cdot \max\{|x|,|y|^{\lambda},|z|\}##

##\textbf{Triangle inequality}##

##|\max\{|x_1|,|y_1|^{\lambda},|z_1|\} +\max\{|x_2|,|y_2|^{\lambda},|z_2|\}| \leq |\max\{|x_1|,|y_1|^{\lambda},|z_1|\}| + |\max\{|x_2|,|y_2|^{\lambda},|z_2|\}|##

Now, I'm not sure, but ##\lambda## can now take on all values without the triangle inequality not being fulfilled. At least I can't think of an example where the triangle inequality is not fulfilled.