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The metrizable linear extensions of metrizable sets in topological linear spaces


Author: L. Drewnowski
Journal: Proc. Amer. Math. Soc. 51 (1975), 323-329
MSC: Primary 46A15
DOI: https://doi.org/10.1090/S0002-9939-1975-0380336-9
MathSciNet review: 0380336
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Abstract: Suppose a subset $ X$ of a Hausdorff [locally convex] topological linear space $ (E,\tau )$ is metrizable in its relative topology $ \tau \vert X$. It is shown that if $ \tau \vert X$ is separable, then there exists a metrizable [locally convex] linear topology $ {\tau _0}$ on the subspace $ V$ generated by $ X$ such that $ {\tau _0} \subset \tau \vert V$ and $ {\tau _0}\vert X = \tau \vert X$ (Theorem 2). This is related to a recent result of Larman and Rogers which states that if, in addition, $ X$ is locally bounded, then $ {\tau _0}$ can be chosen to be normable (but then not necessarily $ {\tau _0} \subset \tau \vert V$) (Theorem 1). It is then observed that $ {\tau _0}\vert X = \tau \vert X$ does not mean the coincidence of the corresponding induced uniformities on $ X$. However, this is the case if the invariant uniformity compatible with $ \tau $ is metrizable on $ X$ and $ X$ is convex (Theorem 4).


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0380336-9
Keywords: Topological linear space, induced topology, second countable topology, metrizable linear topology, norm, induced uniformity, metrizable uniformity, convex set, absolutely convex hull
Article copyright: © Copyright 1975 American Mathematical Society

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