Polishable Subspaces of Infinite-Dimensional Separable Banach Spaces
Maciej Malicki · Real Analysis Exchange · 2008
We show that there exist Polishable subspaces of arbitrarily high Borel class in every infinite-dimensional separable Banach space. Introduction.In this paper we study Polishable subspaces of infinite-dimensional separable Banach spaces, analogous to Polishable subgroups of Polish groups (that is, topological groups whose topology is separable and completely metrizable).It is known [1] that there exist Polishable subgroups of arbitrarily high Borel class in every non-discrete abelian Polish group.A natural question arises whether the same is true of infinite-dimensional separable Banach spaces.We answer this question in the positive by constructing a family of Polishable subspaces in l 1 , in a manner similar to the one presented in [4].The general statement follows from the fact that l 1 can be continuously embedded in every Banach space. Some Background, Notation and Definitions.All Banach spaces considered in this paper are assumed to be separable.By a linear subspace of a Banach space X we mean not only closed subspaces but all subsets of X closed under addition and scalar multiplication.A linear