In some textbooks metric spaces you sometimes encounter these "$l_p$ metrics", $d_1, d_2, d_\infty$

(I don't think $l_p$ metric is very standard usage)

For example,

$d_1(x,y) := \sum\limits_i^m |x_i - y_i|$

$\left\| \boldsymbol{x} \right\| _1 := \sum_{i=1}^{n} \left| x_i \right| $

Is there a general $l_p$ metric and are $l_p$ metrics induced by $l_p$ norms?

  • $\begingroup$ In general, if you have a norm $\|\cdot\|$, you can define a metric that is induced by it by $d(x,y) = \|x-y\|$. This is how we think about distances between points in Euclidean space. You take the vectors, subtract them and measure the length of that vector and that is the distance between them. $\endgroup$ – Cameron Williams Feb 3 '16 at 3:11
  • $\begingroup$ On the notation question, $l_p$ usually denotes a countably infinite-dimensional space, such as a set of infinite sequences. I don't think I've seen it denote a finite-dimensional space. $\endgroup$ – Paul Feb 3 '16 at 15:57

$l_p$ metric on $\mathbb R^m$: $$ d_p(x-y) = \left(\sum_{i=1}^m \big|x_i-y_i\big|^p\right)^{1/p} $$ $1\le p < \infty$. I will let you do the $l_\infty$ metric.

  • $\begingroup$ For the value of $p$, maybe it's more common to write $1\le p\le \infty$? $\endgroup$ – Vim Feb 3 '16 at 3:26
  • $\begingroup$ Yes. Only that formula is not valid for $p=\infty$. You have to use a different formula then. $\endgroup$ – GEdgar Feb 3 '16 at 15:47
  • $\begingroup$ Oh sorry I don't see this point. $\endgroup$ – Vim Feb 3 '16 at 15:50

(some changes in order to improve precision)

About your last interrogation. Any normed vector space defines naturally a metric space by the relationship $d(x,y)=\|x-y\|_p$. Thus, indeed, any $\|\cdot\|_p$ norm induces naturally an $\ell_p$ "metrics" (synonym : "$\ell_p$ distance").

A point of vocabulary about the words "metrics" vs. "distance". "Metrics" is used in applications such as image processing. In mathematics, in differential geometry, it designates a tensor $g_{ij}$ ; topologists prefer "distance" to "metrics".


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