Homeomorphisms between banach spaces

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 is(i) proper (Banach-Mazur) or (ii) ò^¥₀inf_x£sl/(F'(x)]^-1ds = ¥ (Hadamard-Levy), then (L) is satisfied. Other analytic conditions are also given.