Two topological problems concerning infinite-dimensional normed linear spaces

Abstract

Introduction. In the present work, we shall study the following two closely related topological problems concerning infinite-dimensional normed linear spaces. Problem I. Given a closed subset K of an infinite-dimensional normed linear space E, when is E K homeomorphic to E? Problem II. Given a closed subset K of an infinite-dimensional normed linear space E, when does there exist a periodic homeomorphism of E onto E with K as its set of fixed points? Using the fact that every nonreflexive Banach space contains a decreasing sequence of nonempty bounded closed convex subsets with empty intersection, Klee [11] proved that if K is a compact subset of a nonreflexive Banach space E, then E is homeomorphic to E K. Later [13], he showed that every infinite-dimensional normed linear space contains a decreasing sequence of unbounded but linear bounded (for the definition, see ?1) nonempty closed convex subsets with empty intersection. He used this result to prove that every infinite-dimensional normed linear space E is homeomorphic to E K where K is an arbitrary compact subset of E. As a consequence [13], if C is the unit cell of an infinite-dimensional normed linear space E, then there exists a homeomorphism i of C onto a closed half-space J in E such that i (Bd C) = Bd J. Klee [11] also proved that if E is either a nonreflexive strictly convexifiable Banach space or an infinite-dimensional lp-space, then Q is homeomorphic to Q U K where K is a compact convex subset of the bounding hyperplane of an open halfspace Q in E. Concerning the set of fixed points of a periodic homeomorphism of a topological space into itself, a classical result of Smith [19] states that if M is a finite-dimensional locally compact space, acyclic modp where p is