Abstract and Applied Analysis
Volume 2003 (2003), Issue 7, Pages 407-433
Revisiting Cauty's proof of the Schauder conjecture
1Department of Mathematics, Pittsburg State University, Pittsburg 66762, KS, USA
2Department of Mathematics, University of Missouri-Columbia, Columbia 65211, MO, USA
Received 18 April 2002
Copyright © 2003 Tadeusz Dobrowolski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The Schauder conjecture that every compact convex subset of a metric linear space has the fixed-point property was recently established by Cauty (2001). This paper elaborates on Cauty's proof in order to make it more detailed, and therefore more
accessible. Such a detailed analysis allows us to show that the convex compacta in metric linear spaces possess the simplicial approximation property introduced by Kalton, Peck, and Roberts. The latter demonstrates that the original Schauder approach to solve the conjecture is in some sense correctable.