Homeomorphisms Between Banach Spaces

Abstract

We consider the problem of finding precise conditions for a map F between two Banach spaces X, Y to be a global homeomorphism.Using methods from covering space theory we reduce the global homeomorphism problem to one of finding conditions for a local homeomorphism to satisfy the "line lifting property."This property is then shown to be equivalent to a limiting condition which we designate by (L).Thus we finally show that a local homeomorphism is a global homeomorphism if and only if (L) is satisfied.In particular we show that if a local homeomorphism isOther analytic conditions are also given.Introduction.Suppose we have a continuous (or continuously differentiable) map F between two Banach spaces X and Y.We ask what additional assumptions must be imposed upon F to insure that it is a homeomorphism (diffeomorphism) of X onto TI We observe that if F is a homeomorphism then in particular it is a local homeomorphism.Also if F is a diffeomorphism, then by the chain rule F'(x) is an invertible linear map for every x.Since these conditions are necessary for our problem, we shall always assume that our map F is either a local homeomorphism or, if F eC1(Ar), that F'(x) is invertible.Thus we ask when such a map is a homeomorphism (diffeomorphism) of X onto Y.Using Lemma 1.1 we shall see that we can reduce our problem to a more fundamental one: that of determining when a given local homeomorphism F is a covering space map of X onto Y.In § 1 we present the following solution to this question (Theorem 1.1):If F:D Ç X-*-Y is a local homeomorphism (where X and Y are Banach spaces, D open), then (Z), F) is a covering space of FiD) if and only if F "lifts lines."We next introduce a simple analytic criterion, condition (L), which insures