what does mean by $\max_i$ in metric space

Question

what does mean by : $$d(x,y) = \max_i|x_i - y_i|, \quad\text{where} \hspace{0.25cm}x_i,y_i \in \mathbb{R}^n $$ Can some one explain me what does it mean by "$\max_i$"?

(note: I am studying metric space in physics, maybe my question looks trivial)

Answer 1

Generally, the notation $$ \operatorname{Op}_i\text{el}_i $$ means "Apply the operation Op to the operand $\text{el}_i$ for each possible $i$". We can include limitations on the $i$ if it isn't clear from context.

Thus for your case, it means "Take the max of $|x_i-y_i|$ for each possible $i$". Other common examples include

• "$\sum_{n\in \Bbb N} \frac1{n^2+1}$" means "sum $\frac1{n^2+1}$ for each natural $n$"
• "$\bigcup_k X_k$" means "Take the union of all the $X_k$"

Answer 2

Let $x = (x_1, ... , x_n)$ and $y = (y_1, ... , y_n)$ be elements of $\mathbb R^n$. Then

$$\operatorname{max}_i |x_i - y_i|$$

means the largest of the numbers $|x_1 - y_1|, |x_2-y_2|, ... , |x_n - y_n|$.