Spacesnonexpansive

Spacesnonexpansive is not a widely used term in mathematics. It most closely relates to the study of nonexpansive mappings in metric and normed spaces, where a map f: X → X is nonexpansive if d(f(x), f(y)) ≤ d(x, y) for all x, y in X. In a normed space this becomes ||f(x) − f(y)|| ≤ ||x − y||. If the map is linear, nonexpansiveness is equivalent to having operator norm ≤ 1.

The concept centers on how spaces support nonexpansive maps and the fixed-point phenomena that arise there. Examples of nonexpansive maps include the identity map, which is trivially nonexpansive, and the metric projections onto nonempty closed convex subsets of a Hilbert space, which are nonexpansive as well. In Banach spaces and more general metric spaces, the geometry of the space influences which nonexpansive maps have fixed points and how iterative methods converge.

Fixed-point theory provides key results for nonexpansive maps. While nonexpansive self-maps do not in general admit fixed points, certain spaces and conditions guarantee existence. For instance, Browder–Kirk type theorems show that every nonexpansive map from a nonempty closed convex bounded subset of a uniformly convex Banach space into itself has a fixed point. Additional tools include demiclosedness principles and asymptotic regularity, which help establish convergence in iterative schemes. Stronger notions, such as firmly nonexpansive mappings, also play a central role in optimization and numerical analysis.

Because spacesnonexpansive is not a standard term, this article treats it as a reference to the broader framework of nonexpansive mappings and the spaces on which they act. See also nonexpansive map, fixed-point theory, proximal mappings, and CAT(0) spaces.