 Triangle inequality

A metric satisfies $d(x,z)\le d(x,y)+d(y,z)$. A norm satisfies the corresponding inequality $\Vert u+v\Vert \le \Vert u\Vert +\Vert v\Vert$.

The triangle inequality is a fundamental concept in geometry and mathematics that states the following for any triangle with sides of lengths \( a \), \( b \), and \( c \): 1. \( a + b > c \) 2. \( a + c > b \) 3. \( b + c > a \) In essence, the triangle inequality asserts that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.