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The article has too narrow of definition of what a norm is--since around 1910 or so, we've had to have a more expansive notion of what can be a norm. e.g. the
by rhelz 2y ago
The article has too narrow of definition of what a norm is--since around 1910 or so, we've had to have a more expansive notion of what can be a norm.
e.g. the norm for Minkowski's space-time can be negative and does not satisfy the triangle inequality.
Imagine the sheer courage it took for Minkowski to propose a distance function which did not satisfy the triangle inequality.....the moral of the story is yes, learn up on these concepts, but don't be so wedded to somebody else's definition that you can't see when it should be relaxed and generalized.
- Tainnor 2y agoDefinitions are generalized all the time in mathematics. If Minkowski discovered something that was similar to a norm but not quite, that may be interesting, but it's still not a norm. It also has nothing to do with the complex numbers where the canonical norm is, in fact, a norm.
- rhelz 2y agoThe Minkowski Norm is, in fact, a norm: https://en.wikipedia.org/wiki/Minkowski_norm https://en.wikipedia.org/wiki/Minkowski_norm // It also has nothing to do with the complex numbers // The author of the linked article said that what he was describing was a norm because it satisfied the triangle inequality (among other things). I'm just pointing out that the (among other things) was enough for it to be a norm. I certainly wished somebody had pointed out to me that not all norms need to satisfy the triangle property. There have been spaces where I wasted months of effort trying to find a norm which satisfied the triangle inequality, when a more minkowski-like norm would have been perfectly adequate.
- Tainnor 2y agoYou're playing semantic games. If it doesn't satisfy the triangle inequality it's a quasi-nonorm. When you say "norm", people understand it to imply the triangle inequality. That doesn't mean that spaces without a proper norm are useless. Also your link links to a disambiguation page.