triangle inequality

the triangle inequality

1. (mathematical analysis) The inequality that states that the magnitude of the sum of two vectors is less than or equal to the sum of the magnitudes of the vectors.

   ${\displaystyle \|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|}$

2. (mathematical analysis) The inequality that states that the distance between two points is less than or equal to the sum of the distances between the first two points and some third point.

   ${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$

3. (mathematical analysis, by abstraction) Either of the above two inequalities, considered as one of the axioms which a function must satisfy to be a norm or metric respectively.