The metrizable linear extensions of metrizable sets in topological linear spaces

Abstract:

Suppose a subset $X$ of a Hausdorff [locally convex] topological linear space $(E,\tau )$ is metrizable in its relative topology $\tau |X$. It is shown that if $\tau |X$ is separable, then there exists a metrizable [locally convex] linear topology ${\tau _0}$ on the subspace $V$ generated by $X$ such that ${\tau _0} \subset \tau |V$ and ${\tau _0}|X = \tau |X$ (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 ${\tau _0} \subset \tau |V$) (Theorem 1). It is then observed that ${\tau _0}|X = \tau |X$ 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).