We look for characterizations of those locally convex spaces that satisfy the strict MACKEY convergence condition within the context of spaces with webs. We will say that a locally convex space has a boundedly compatible web if it has a web of absolutely convex sets whose members behave like zero neighborhoods in a metrizable locally convex space. It will be shown that these locally convex spaces satisfy the strict MACKEY convergence condition. One consequence of this result will be a characterization of boundedly retractive inductive limits. We will also prove that if E is locally complete and webbed, then the strict MACKEY convergence condition is equivalent to E having a boundedly compatible web.
IntroductionIf a HAUSDORFF locally convex space satisfies the strict MACKEY convergence condition, then it possesses several desirable features, such as the metrizability of each of its bounded subsets, convergence of sequences with respect to a normed topology, and the satisfaction of the strong dual density condition of Bierstedt and BONET [l]. Because rnetrizable locally convex spaces are the simplest examples that satisfy the strict MACKEY convergence condition, it makes sense to use them as a model. This will be our approach, within the more general context of locally convex spaces with webs. Detailed discussions of webs may be found in [2], [7], [lo], and [ll]. The strategy will be to look for situations where the members of the web mimic the zero neighborhoods of a metrizable space. Such spaces will be labeled as having a boundedly compatible web, and we will prove that they satisfy the strict MACKEY convergence condition. It will be easy to see that every metrizable locally convex space has a boundedly compatible web, but more will be obtained. We will show that a locally convex inductive limit of spaces with boundedly compatible webs is boundedly retractive precisely when the inductive limit again has a boundedly compatible web. Moreover, it will turn out that boundedly compatible webs fit nicely into the framework of so-called sequentially webbed (and quasi-sequentially webbed) spaces of [3] and [4]. Next, we would like to have conditions that ensure a converse. Our basic charac-
;. The problem of characterizing those locally convex spaces satisfying the Mackey convergence condition is still open. Recently in [4], a partial description was given using compatible webs. In this paper, those results are extended by using quasi-sequentially webbed spaces (see Definition 1). In particular, it is shown that strictly barrelled spaces satisfy the Mackey convergence condition and that they are properly contained in the set of quasi-sequentially webbed spaces. A related problem is that of characterizing those locally convex spaces satisfying the socalled fast convergence condition. A partial solution to this problem is obtained. Several examples are given.
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