Ultrametric Space - Formal Definition

Formal Definition

Formally, an ultrametric space is a set of points with an associated distance function (also called a metric)

(where is the set of real numbers), such that for all, one has:

1. iff
2. (symmetry)
3. (strong triangle or ultrametric inequality).

The last property can be made stronger using the Krull sharpening to:

with equality if

For simplicity, let us use the norms instead of the distances in the proof; we want to prove that if, then the equality occurs if . Without loss of generality, let's assume that . This implies that . But we can also compute . Now, the value of cannot be, for if that is the case, we have contrary to the initial assumption. Thus, and . Using the initial inequality, we have and therefore .