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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Domain invariance in infinite-dimensional linear spaces


Author: Jan van Mill
Journal: Proc. Amer. Math. Soc. 101 (1987), 173-180
MSC: Primary 57N17; Secondary 46B99, 57N20
DOI: https://doi.org/10.1090/S0002-9939-1987-0897091-4
MathSciNet review: 897091
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Abstract: Let $ X$ be an infinite-dimensional locally convex linear space. It is known that $ X$ is homeomorphic to a subspace of $ X$ which is not open. We prove that every Banach space $ B$ contains a dense linear subspace $ L$ with the following property: If $ U \subseteq L$ is open and if $ f:U \to L$ is continuous and injective, then there exists a dense open $ V \subseteq U$ such that (a) \begin{displaymath}\begin{gathered}f(V) \end{gathered} \end{displaymath} is open in \begin{displaymath}\begin{gathered}L \end{gathered} \end{displaymath}, and (b) $ f:V \to f(V)$ is a homeomorphism. As a consequence, $ L$ and $ L \times {\mathbf{R}}$ are not homeomorphic.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0897091-4
Keywords: Normed linear space, domain invariance
Article copyright: © Copyright 1987 American Mathematical Society