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 is (i) proper (Banach-Mazur) or (ii) $\int _0^\infty {{{\inf }_{||x|| \leqslant s}}} 1/||{[F’(x)]^{ - 1}}||ds = \infty$ (Hadamard-Levy), then $(L)$ is satisfied. Other analytic conditions are also given.