TITLE: Motivation for triangle inequality
QUESTION [18 upvotes]: Triangle inequality is used in one context or the other in analysis.
To list a few
$$ \|x+y\| \leq \|x\| + \|y\| $$
$$ d(x,y) \leq d(x,z) + d(z,y) $$
$$ \mu(A \cup B) \leq \mu(A) + \mu(B) $$
What is the motivation for imposing this? Intutively, it feels if we do not impose triangle inequality, the space will collapse to a trivial space in some sense. I am not able to express this formally. I would appreciate if someone could could throw some light on this and the motivation for triangle inequality?

REPLY [7 votes]: It seems I can't add comments so I decided to post this as an answer, I apologize if it seems trivial or redundant. My point of view is that most occurrences of this inequalities are just minimum principles (something you could summarize as "min $\le$ any"). In each case, the quantity (distance, norm, dimension, cardinal...) can be viewed as a minimum :

The distance between two points is the length of the shortest path between them
The dimension of a vector space smallest size for a system of generator
The cardinal of a set is the smallest $n$ for which your set can be embedded in $[\!|1,n|\!]$

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In that setting, the triangle inequality more or less states that the right hand side correspond to a possibility (of path, generating set, embedding...) :

A path from $x$ to $z$ and then $z$ to $y$ is a path from $x$ to $y$.
The union of a generating sets gives a generating set for the sum.
If you can embed $A$ in $A'$ and $B$ in $B'$ then $A \cup B$ can be embedded in the disjoint union of $A'$ and $B'$ 

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