Lipschitzbounded

Lipschitzbounded (often written Lipschitz-bounded) describes a property of a collection of functions whose rates of change are uniformly controlled by a single constant. In metric space terms, let X and Y be metric spaces with distances d_X and d_Y. A function f: X → Y is Lipschitz with constant L ≥ 0 if for all x1, x2 ∈ X, d_Y(f(x1), f(x2)) ≤ L d_X(x1, x2). A family F of functions from X to Y is called Lipschitz-bounded if there exists a constant L such that every f ∈ F is L-Lipschitz. Some contexts also require a uniform bound on the functions themselves, yielding a bound in the corresponding Lipschitz norm.

Examples: The set of all functions f(x) = ax + b from the real line to itself with |a| ≤ L is L-Lipschitz-bounded. The unit ball in the Lipschitz function space Lip(X, Y), consisting of all functions with Lipschitz constant at most 1, is another instance of a Lipschitz-bounded collection.

Properties: A Lipschitz-bounded family is automatically equicontinuous on X, since a single L controls how far the function values can change. This use of a uniform Lipschitz bound underpins many compactness criteria, such as those in the Arzelà–Ascoli theorem, when X is compact and the family is additionally pointwise bounded.

Applications: Lipschitz-boundedness helps in analysis and approximation theory to bound rates of change, to study function spaces, and to establish compactness and convergence results in various settings.

See also: Lipschitz continuity, Lipschitz constant, equicontinuity, Arzelà–Ascoli theorem, Banach spaces.