The Banach fixed point theorem, also called the Banach contraction principle, guarantees a unique fixed point for a contraction
mapping of a nonempty closed subset of a Banach space
into itself. Let 
be a contraction mapping
from a nonempty closed subset 
of a Banach space 
into 
. Then there exists a unique 
such that 
.
Banach Fixed Point Theorem
See also
Contraction Mapping, Fixed Point TheoremExplore with Wolfram|Alpha
References
Debnath, L. and Mikusiński, P. Introduction to Hilbert Spaces with Applications. Boston, MA: Academic Press, 1990.Referenced on Wolfram|Alpha
Banach Fixed Point TheoremCite this as:
Weisstein, Eric W. "Banach Fixed Point Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BanachFixedPointTheorem.html