Easy question about sup norm

Question

Im reading through terence tao's Analysis2 book and came across the following page describing the sup norm metric.

I dont understand the example at the end. The difference $|x_i - y_i|$ refers to the $i$-th index of in this case $(1,6)$ and $(4,2)$. Why does this not result in the $sup(|1-4|=3,|6-2|=4)$. And second, why in the example is the supremum of (5,2) equal to 7 instead of the maximum of the numbers 5 and 2, which would be 5?

Answer 1

Yes, this is clearly a mistake. The first one is taking $d_{l^\infty}((1,4), (6,2) = \max\{5,2\}$ instead of $d_{l^\infty}((1,6), (4,2) = \max\{3,4\}$ and the second one is not actually taking the Maximum, but the sum, which would be the $l_1$ norm.

I guess they wanted to give an example of a different norm than the euclidian norm, but they mixed up $l_1$ and $l_\infty$ in the process.

Answer 2

Your are right, this is an error. Instead it should be $$ d_{l^\infty}((1,6),(4,2))=\sup\{3,4\}=4. $$ This error has already been recorded in the list of errata, which you can find here: https://terrytao.wordpress.com/books/analysis-ii/