Retakh's conditions and regularity properties of (LF)-spaces
A countable inductive limit \(E=\text{ind }E_n\) is said to be sequentially retractive if for each null sequence \((x_k)_{k\in \mathbb{N}}\) in \(E\), there exists \(n=n(x_k)\in \mathbb{N}\) such that \((x_k)_{k\in\mathbb{N}}\) is contained and a null sequence in \(E_n\) [\textit{K. Floret}, J. Reine Angew. Math. 259, 65-85 (1973; Zbl 0251.46003)]. In the present paper, a countable inductive limit \(E=\text{ind }E_n\) is said to be weakly sequentially retractive if for each weak null sequence \((x_k)_{k\in\mathbb{N}}\) in \(E\), there exists \(n=n(_k)\in\mathbb{N}\) such that \((x_k)_{k\in\mathbb{N}}\) is contained and a weak null sequence in \(E_n\). It is proved that weak sequential retractivity implies regularity. As is well-known, an (LF)-space satisfying Retakh's condition \((M_0)\) needn't be regular [\textit{D. Vogt}, North-Holland Math. Stud. 170, 57-84 (1992; Zbl 0779.46005)]. For (LF)-spaces \(E=\text{ind }E_n\) with Retakh's condition \((M_0)\), a characteristic condition of regularity is given. That is, for each \(n\in\mathbb{N}\) there is \(m=m(n)\geq n\) such that for every bounded set \(B\) in \(E_n\), \(\overline B^{E_k}\subset E_m\) for all \(k\geq n\). In fact, for those (LF)-spaces, \[ \alpha\text{-regularity}\Leftrightarrow\text{regularity}\Leftrightarrow\text{ weak sequential retractivity}. \] As a corollary, it is proved that for countable inductive limits of weakly sequentially complete Fréchet spaces, Retakh's condition \((M_0)\) implies weak sequential completeness, weak sequential retractivity and regularity.
weakly sequentially complete Fréchet spaces
scientific article; zbMATH DE number 938607