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SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL
Published online by Cambridge University Press: 07 August 2017
Abstract
For a Tychonoff space $X$, let
$\mathbb{V}(X)$ be the free topological vector space over
$X$,
$A(X)$ the free abelian topological group over
$X$ and
$\mathbb{I}$ the unit interval with its usual topology. It is proved here that if
$X$ is a subspace of
$\mathbb{I}$, then the following are equivalent:
$\mathbb{V}(X)$ can be embedded in
$\mathbb{V}(\mathbb{I})$ as a topological vector subspace;
$A(X)$ can be embedded in
$A(\mathbb{I})$ as a topological subgroup;
$X$ is locally compact.
Keywords
MSC classification
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 97 , Issue 1 , February 2018 , pp. 110 - 118
- Copyright
- © 2017 Australian Mathematical Publishing Association Inc.
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