Let 
be a closed convex subset of a Banach space and assume
there exists a continuous map 
sending 
to a countably compact subset 
of 
. Then 
has fixed points.
Schauder Fixed Point Theorem
See also
Map Fixed PointExplore with Wolfram|Alpha
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 543, 1980.Schauder, J. "Der Fixpunktsatz in Funktionalräumen." Studia Math. 2, 171-180, 1930.Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995.Referenced on Wolfram|Alpha
Schauder Fixed Point TheoremCite this as:
Weisstein, Eric W. "Schauder Fixed Point Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchauderFixedPointTheorem.html