Convex bodies and periodic homeomorphisms in Hilbert space

Abstract

Introduction. We are going to discuss certain topological properties of infinite-dimensional normed linear spaces, the results being most complete for Hilbert space 'i. Let us begin by describing some results and questions of previous authors which are closely related to those of the present paper. From A. Tychonoff's fixed-point theorem ([31], 1935)(2) it follows that in the weak topology, the unit cell C= {x| |x|| < 1 } of 'i must have the fixedpoint property; that is, every weakly continuous map of C into itself admits at least one fixed point. In the norm topology, on the other hand, S. Kakutani ([13], 1943) described a homeomorphism without fixed points of C onto itself. He used this to show that the unit sphere S ={x x = 1 } is contractible and is a deformation retract of C. At the end of his paper, Kakutani raised several questions. Are any two of 'i, C, and S homeomorphic? Does C admit a periodic homeomorphism without fixed points? What is the situation in general Banach spaces? In partial answer to the last question, J. Dugundji ([7], 1951) proved that the unit cell of a normed linear space has the fixed point property only if the space is finite-dimensional. P. A. Smith had proved ([26], 1941) that each prime-period homeomorphism of Euclidean n-space En must have a fixed point and asked ([9, p. 259], 1949) whether 6i admits a period two homeomorphism without fixed points. 0. H. Keller ([14], 1931) proved that the infinite-dimensional compact convex subsets of are mutually homeomorphic and all homogeneous. W. A. Blankinship ([3], 1951) showed that if XC, and Cl X is compact, then i-i-X is contractible. In the present paper we answer the questions of Kakutani and Smith, strengthen the theorems of Keller and Blankinship, and establish some further topological properties of convex bodies and periodic homeomorphisms in 'i. The principal tools employed are (a) Mazur's homeomorphism [22] of the space (L') onto the space (L2); (b) the existence in every nonreflexive normed linear space of a decreasing sequence of bounded closed convex sets with empty intersection; (c) the existence in (L') of a one-parameter con-