Mackey convergence and quasi-sequentially webbed spaces
The previous paper of the author [Intern. Math. Math. Sci. 11, No. 3, 473-483 (1988; Zbl 0664.46003)] presented a partial solution of the characterization of locally convex spaces with Mackey convergence condition, using a technique of compatible webs. The results of the paper are applied to the quasi-sequentially webbed spaces (a definition of the sequentially webbed space refers to a weaker condition on strands than for sequentially webbed spaces, namely, for each null sequence it is required that members of the sequence are eventually contained in a closed absolutely convex hull of the strands). The author proves then that every metrizable lcs is a quasi-sequentially webbed locally convex space with Mackey convergence condition (MCC) (the previous results of the author were formulated for locally Baire and webbed locally convex spaces); and the class of quasi-sequentially webbed spaces is stable under taking subspaces, sequentially retractive inductive limit, etc. Also, every quasi-sequentially webbed space satisfies the Mackey convergence condition. The author proves also that if the lcs \(E\) is webbed and locally Baire (i.e. for each bounded set \(A\) there exists a bounded disk \(B\) in \(A\) such that \(E_ B\) is a Baire space), and satisfies the Mackey convergence condition, then \(E\) is quasi-sequentially webbed. The rest of the paper is devoted to the fast convergence condition [see \textit{H. Jarchow} and \textit{J. Swart}, Isr. J. Math. 16, 150-158 (1973; Zbl 0272.46002)]. In particular, the following theorem is proved: The following three conditions are equivalent for a locally convex space \(E\): 1) \(E\) satisfies the fast convergence condition, 2) for each null-sequence there exists a compact disc \(K\) such that the sequence converges in \(E_ K\), 3) \(E\) is locally complete MCC.
characterization of locally convex spaces with Mackey convergence condition
scientific article; zbMATH DE number 7763