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Keywords:
inductive limits; regularity; sequential retractivities
inductive limits; regularity; sequential retractivities
Summary:
In this paper we prove the following result: an inductive limit $(E,t) = \text{ind}(E_n,t_n)$ is regular if and only if for each Mackey null sequence $(x_k)$ in $(E,t)$ there exists $n=n(x_k)\in \mathbb N$ such that $(x_k)$ is contained and bounded in $(E_n,t_n)$. From this we obtain a number of equivalent descriptions of regularity.
References:
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[2] K. Floret: Folgenretraktive Sequenzen lokalkonvexer Räume. J. Reine Angew. Math. 259 (1973), 65–85. MR 0313748 | Zbl 0251.46003
[3] Q. Jing-Hui: Retakh’s conditions and regularity properties of (LF)-spaces. Arch. Math. 67 (1996), 302–307. DOI 10.1007/BF01197594 | MR 1407333 | Zbl 0858.46007
[4] J. Bonet and P. Perez Carreras: Barrelled locally convex spaces. North-Holland Math. Stud. 131, Amsterdam, 1987. MR 0880207
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