4
$\begingroup$

The production of two metrics is a metric also. It's googled easy. But what's about a sum? As I can see sum is metric, as the triangle inequality of metric sum is the consequence of the inequality feature and two other axioms looks obvious. Is it correct?

$\endgroup$
  • 1
    $\begingroup$ yes, it is. <<characters limit for a comment>> $\endgroup$ – mm-aops Jun 6 '14 at 21:07
  • 1
    $\begingroup$ Indeed, google is not a theorem prover. $\endgroup$ – Ruslan Jun 6 '14 at 21:12
  • 1
    $\begingroup$ @nomen I try and described my view. $\endgroup$ – SerG Jun 6 '14 at 21:17
  • 1
    $\begingroup$ You have in essence given a proof. $\endgroup$ – André Nicolas Jun 6 '14 at 21:20
  • 1
    $\begingroup$ The product of metrics need not be a metric in general. The sum always is. $\endgroup$ – Henno Brandsma Sep 25 '18 at 4:49
5
$\begingroup$

In short, yes. Your sketched argument is correct for the triangle inequality (we're just adding two valid inequalities to get a third). One does also need to remark that the sum of two numbers $\ge 0$ can only be $0$ if both summands are $0$ (and then we apply the axiom for the two constituent metrics after that).

$\endgroup$

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.