Distance of two vectors under L_inf norm

Question

Very simple question: suppose I have two vectors $a = (1,-2)$ and $b = (4,2)$. Under the L_inf norm, would the distance between them be $abs( ||a||_{inf} - ||b||_{inf}) = abs(2 - 4) = 2$? Is this the correct way of measuring distance under a norm?

Answer 1

No, you apply the norm (in our case $\| \cdot\|_\infty$) to the vector difference to get the distance: $$\|a - b\|_\infty = \|(1-4,-2-2)\|_\infty = \max \{|-3|,|-4| \} = 4$$