I'm doing a course on metric spaces at the moment, and I've just come across the product metrics $d_1, d_2, d_\infty$, which are defined in the following way for metric spaces $(X, d_X), (Y, d_Y)$ and $x_1, x_2 \in X$, $y_1, y_2 \in Y$:

$$d_1((x_1, y_1), (x_2, y_2)) := d_X(x_1, x_2) + d_Y(y_1, y_2)$$

$$d_2((x_1, y_1), (x_2, y_2)) := (d_X(x_1, x_2)^2 + d_Y(y_1, y_2)^2)^{1/2}$$

$$d_\infty((x_1, y_1), (x_2, y_2)) := max\{d_X(x_1, x_2), d_Y(y_1, y_2)\}.$$

I have seen these names used in several textbooks on metric spaces as well as in the lectures and notes of the course itself; what is the meaning behind the naming convention $d_1, d_2, d_\infty$?

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    $\begingroup$ By the way, you should have $d_Y (\color{red}{y_1}, y_2)$, right? $\endgroup$ – Eff Oct 7 '16 at 15:05
  • $\begingroup$ @Eff I should have, thank you; fixed. $\endgroup$ – Nethesis Oct 7 '16 at 15:06

In general you have that $$d_p \big((x_1,y_1), (x_2,y_2)\big):= \Big[d_X(x_1,x_2)^p+d_Y(y_1,y_2)^p \Big]^{1/p}.$$ This is in metric spaces, but it is widely used in normed spaces known as $L^p$-norm. You will notice that this fits with the first two, however the last one is slightly different as one uses $p =\infty$. Here one can show that $\|\cdot\|_\infty$-norm is the limit of $\|\cdot\|_p$ as $p\to\infty$.

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  • $\begingroup$ Great, thank you! I was wondering why it seemed to jump so quickly. $\endgroup$ – Nethesis Oct 7 '16 at 15:00

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