JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 97, 301-305 (1983) A Proof of the Ryll-Nardzewski Fixed Point Theorem JAMES DUGUNDJI Mathematics Department, University of Southern California, Los Angeles, California 90007 AND ANDRZEJ GRANAS * Dkpartement des Mathe’matiques, UniversitP de MontrPal, MontrPal, QuPbec, Canada Submitted by K. Fan If .F is any family of maps of a space X into itself, by a fixed point for ,7 is meant a point x0 E X such that T(x,) = x0 for each T E F. Our aim in this note is to present a self-contained proof of the theorem of Ryll- Nardzewski on fixed points of families of affine self-maps of convex sets. The proof, which clarifies and simplifies some of the arguments in 121, is based entirely on two general results in functional analysis: the extended Krein-Milman theorem and the Mazur theorem in locally convex spaces. We begin with DEFINITION 1. Let .F be a family of self-maps of a space X. The family .F is called distal on X if for each pair x f y of distinct points of X, there is an open covering { I’, / a E 67) of X such that T( y> hf U ( V, 1T(x) E V, } for each TE.7. The requirement of this definition, that for no T E .F do TX and Ty belong to a common set V,, can be stated directly in terms of the Cartesian product X x X: for x # y, the set {(TX, Ty) 1 T E ,F) must be outside some neighbourhood 0 (V, x V, / a E c?) of the diagonal; the neighbourhood depends on the pair x, y and the notion of distal clearly depends on the topology in X. If. 7 is a family of self-maps of X, a subset A c X is called .F-invariant if T(A) CA for all T E ,F; a closed A CX which is .F-invariant and has no * This research was supported in part by a grant from the National Research Council of Canada. 301 0022-247X/83 $3.00 CopyrIght 0 1983 by Academic Press, Inc. All rights of reproduction m any form reserved.