metricnotation

Metric notation refers to the conventional symbols and conventions used to denote distance-like functions in mathematics and related fields. A metric on a set X is a function d: X×X → [0,∞) that satisfies positivity, identity of indiscernibles, symmetry, and the triangle inequality. In practice, d is usually denoted by d, with subscripts to indicate the domain, for example d_X on a metric space X. Convergence of sequences is often written x_n → x and is interpreted via d(x_n, x) → 0. A common specialization is norm-induced metrics: d(x,y) = ||x − y||, where ||·|| is a norm.

Examples include Euclidean distance d_E(u,v) = sqrt(sum (u_i - v_i)^2), Manhattan distance d_1(u,v) = sum |u_i - v_i|, Chebyshev distance d_∞(u,v) = max_i |u_i - v_i|, and graph or discrete metrics such as Hamming distance. In calculus and geometry, metric notation extends to metric tensors in differential geometry, where the line element ds^2 = g_ij dx^i dx^j uses a metric g to determine infinitesimal distances on a manifold.

Geometrically, a metric induces a topology and notions of open balls B(x,r) = {y: d(x,y) < r}. The term metric is distinct from a norm, though a norm on a vector space defines a metric via d(x,y) = ||x − y||.

Applications span analysis, topology, and computer science, including clustering, nearest-neighbor search, and error-correcting distances like Hamming distance. The notation is used consistently across disciplines, but details vary by author and field.