Triangle Inequality - Normed Vector Space

Normed Vector Space

In a normed vector space V, one of the defining properties of the norm is the triangle inequality:

that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.

If the normed space is euclidean, or, more generally, strictly convex, then if and only if the triangle formed by ,, and, is degenerate, that is, and are on the same ray, i.e., or, or for some . This property characterizes strictly convex normed spaces such as the spaces . However, there are normed spaces in which this is not true. For instance, consider the plane with the norm (the Manhattan distance) and denote and . Then the triangle formed by ,, and, is non-degenerate but