nLab Banach fixed-point theorem

Redirected from "Banach fixed point theorem".
Contents

Context

Analysis

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Statement

The Banach fixed-point theorem or contraction mapping theorem states:

Let (X,ρ)(X, \rho) be a sequentially Cauchy complete metric space with a point x 0:Xx_0:X and a rational number C:C : \mathbb{Q} such that for all x:Xx:X and y:Xy:X, ρ(x,y)C\rho(x, y) \leq C. Let T:XXT : X \to X be an endomap with a rational Lipschitz constant 0<c<10 \lt c \lt 1. Then XX has a unique fixed point, a point xx with ρ(T(x),x)=0\rho(T (x), x) = 0, such that for any y:Xy : X with ρ(T(y),y)=0\rho(T (y), y) = 0, x=yx = y.

References

  • Auke B. Booij, Analysis in univalent type theory (pdf)

See also:

Last revised on June 3, 2022 at 07:42:14. See the history of this page for a list of all contributions to it.