Banach fixed-point theorem

The Banach fixed-point theorem states that a contraction on a complete metric space has exactly one fixed point. Repeatedly applying the contraction from any starting point converges to that point.

Contraction mapping: A map between metric spaces that decreases distances by at least a fixed factor less than one. The theorem's central hypothesis is that the map strictly contracts distances.
Picard–Lindelöf theorem: A theorem guaranteeing local existence and uniqueness for solutions of ordinary differential equations with a Lipschitz right-hand side. Its proof casts the differential equation as a fixed-point problem on a function space.
Metric space: A set equipped with a distance function satisfying nonnegativity, symmetry, and the triangle inequality. The theorem requires distances to define contraction and convergence.
Brouwer fixed-point theorem: A theorem stating that every continuous map from a compact convex subset of Euclidean space to itself has a fixed point. It guarantees existence by topology, without requiring a distance-shrinking map or uniqueness.
Complete metric space: A metric space in which every Cauchy sequence converges to a point in the space. Completeness keeps the iterates' limit inside the domain where the fixed point must lie.
Banach's theorem in dynamic programming: The use of contraction mappings to establish a unique value function and convergence of value iteration. Discounting makes the Bellman operator a contraction under standard conditions.
Cauchy sequence: A sequence whose terms become arbitrarily close to one another as the indices increase. The iterates form a Cauchy sequence because successive gaps shrink geometrically.
Schauder fixed-point theorem: A theorem guaranteeing a fixed point for a continuous compact self-map of a suitable closed convex subset of a Banach space. It can handle maps that are not contractions, but does not generally ensure uniqueness.
Fixed point: A point that a function maps to itself. The theorem guarantees existence and uniqueness of this kind of point.
Newton's method: An iterative root-finding method that uses function values and derivatives to update approximations. Unlike contraction iteration, its convergence is generally local and depends on the starting point.

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