Homeomorphisms between banach spaces

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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) ò¥0infx£sl/(F'(x)]-1ds = ¥ (Hadamard-Levy), then (L) is satisfied. Other analytic conditions are also given.

Original languageEnglish (US)
Pages (from-to)169-183
Number of pages15
JournalTransactions of the American Mathematical Society
Volume200
DOIs
StatePublished - 1974
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Keywords

  • Covering space

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