Bornée

Bornée is the feminine form of the French adjective borné, meaning bounded. In mathematics and related fields, it denotes a property of a set, a function, a sequence, or an operator that is contained within finite limits.

In metric spaces, a subset S of a metric space (X, d) is bornée if there exists a point x0 in X and a finite radius R such that d(x, x0) ≤ R for all x in S. Equivalently, S has finite diameter: sup{d(x, y) : x, y ∈ S} < ∞. In Euclidean space R^n this means S is contained in some ball of finite radius. A sequence (x_n) is bornée if there exists M such that |x_n| ≤ M for all n. A function f: D → R is called bornée if there exists M with |f(x)| ≤ M for all x in its domain D.

Examples: The interval [-3, 2] is bornée. The sequence x_n = (-1)^n n is not bornée. The constant function f(x) = 7 is bornée on any domain. The function f(x) = x on R is not bornée.

Etymology and usage: The term derives from borne meaning boundary or limit, with the feminine suffix -ée. In standard French mathematical language it pairs with non bornée for unbounded, and with expressions such as fonction bornée, suite bornée, ensemble borné, and opérateur borné.