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 $d_1, d_2, d_\infty$?

  • 1
    $\begingroup$ 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 $p\to\infty$.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.