Avståndsmetriken

Avståndsmetriken, or distance metric in English, is a fundamental concept in mathematics, particularly in the fields of topology and analysis. It is a function that defines a distance between any two elements of a set. For a set X, a distance metric d is a function d: X × X → ℝ such that for any x, y, and z in X, the following properties hold: non-negativity (d(x, y) ≥ 0), identity of indiscernibles (d(x, y) = 0 if and only if x = y), symmetry (d(x, y) = d(y, x)), and the triangle inequality (d(x, z) ≤ d(x, y) + d(y, z)). These properties ensure that the function behaves intuitively like a measure of distance.

A set equipped with a distance metric is called a metric space. Metric spaces provide a rigorous framework for studying concepts like convergence, continuity, and completeness. Examples of common distance metrics include the Euclidean distance in Euclidean space, the Manhattan distance (or taxicab distance) where movement is restricted to grid lines, and the Hamming distance used for comparing strings of equal length. The choice of distance metric can significantly influence the properties and behavior of a metric space, leading to different topological structures even on the same underlying set. The concept of avståndsmetriken is crucial for defining geometric concepts and understanding the structure of various mathematical spaces.