On the Closed Hull of Subsets of Inductive Limits

Abstract

The present note investigates the closed hull of subsets M contained in the (unrestricted) inductive limit E[τ] of topological vector spaces E_α[τ_α], α∈A. Sufficient conditions for the validity of \bar{M}^τ=∪\limits_α∈A\overline{M∩E_α^τ_α} 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. Futhermore, some inaccuracies contained in some of the standard text books are corrected.