Chapter 22: Nonexpansive Mappings
Heinz H. Bauschke, Walaa M. Moursi · Society for Industrial and Applied Mathematics eBooks · 2023
A mapping T from a subset D of X to Y is Lipschitz continuous with constant L ≥ 0, or simply L-Lipschitz, if ||T(x)−T(y)||Y ≤ L||x−y ||X for all x, y in D, where ||·||X denotes a norm on X and ||·||Y denotes a norm on Y and L ≥ 0. We will often write Tx instead of T(x), etc. If L = 1, we say that T is nonexpansive, while if L < 1, we say that T is a (Banach) contraction. If Y = ℝ, we use ||·||Y = |·|. If we don't single the norm out, the Euclidean norm is used.