Schauder Fixed Point Theorem

Let $A$ be a closed convex subset of a Banach Space and assume there exists a continuous Map $T$ sending $A$ to a countably compact subset $T(A)$ of $A$. Then $T$ has fixed points.

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.