I am wondering why the definition of metric includes 'the triangle inequality'. I browsed some answers before but none of them satisfy me.
I guess it may come from the geometric intuition of n-dimensional Euclidean space because any finite metric space can be isometrically embedded in an Euclidean space as some kind of graph representation of the set itself. This explains why all the finite metric spaces have the same nature: because their metrics can all be interpreted as some 'hidden' Euclidean metric.
This explanation seems to be invalid for infinite metric spaces, because in these spaces I cannot define clearly what an 'infinite' Euclidean space is. However, my intuition still tells me that this kind of explanation might work even in this situation.
The definition of 'norm' adopts the triangle inequality, too, which is a similar question.
I am asking for your help.
Can any infinite metric space be isometrically embedded in some quasi-Euclidean space
definitionfunctional-analysismetric-spaces
Best Answer
The metric function on a metric space intuitively measures the shortest distance between any two points of the space. The triangle inequality has a very simple intuitive meaning: This shortest distance between two points shouldn't shrink if we go via a third point. That would violate how "shortest distance" ought to work.
Euclidean spaces (with some standard metric, like Pythagoras) are the archetypal examples of metric spaces, and one could argue that the notion of metric space is intended to be a generalisation of the notion of Euclidean space (I don't know the actual history here, but it seems reasonable). But it is just that: a generalisation. There are many things that metric spaces can do that Euclidean spaces can't. This is what makes it a useful generalisation: we lose some specificity, so we can't prove all the same things. But on the other hand, the results we do find apply to many non-Euclidean contexts.
Ultimately, that's the reason for the triangle inequality: it works to make a useful generalisation. We could forgo it, and have much more general spaces. This way our results would apply much more broadly. On the other hand, we wouldn't get as many proven results. This is always a trade-off. And the particular trade-off we have made the way we have defined metric spaces has been popular and fruitful.