On the closed hull of subsets of inductive limits (Q1187168)

Summary: The present note investigates the closed hull of subsets \(M\) contained in the (unrestricted) inductive limit \(E[\tau]\) of topological vector spaces \(E_ \alpha[\tau_ \alpha]\), \(\alpha\in A\). Sufficient conditions for the validity of \(\overline{M}^ \tau=\bigcup_{\alpha\in A}\overline{M\cap E_ \alpha}^{\tau_ \alpha}\) are given. These conditions are discussed by examples, and it is shown that there is not any redundance among them. Applications to \(F\)-linked topologies and sequence spaces are indicated. Furthermore, some inaccuracies contained in some of the standard text books are corrected.