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Transactions of the American Mathematical Society

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Homeomorphisms between Banach spaces

Author: Roy Plastock
Journal: Trans. Amer. Math. Soc. 200 (1974), 169-183
MSC: Primary 58C15; Secondary 57A20
MathSciNet review: 0356122
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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 }_{\vert\vert x\vert\vert \leqslant s}}} 1/\vert\vert{[F'(x)]^{ - 1}}\vert\vert ds = \infty $ (Hadamard-Levy), then $ (L)$ is satisfied. Other analytic conditions are also given.

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Keywords: Covering space
Article copyright: © Copyright 1974 American Mathematical Society