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Metrizability of precompact subsets in (LF)-spaces. (English) Zbl 0622.46005

We prove in this paper that every precompact subset in any (LF)-space has a metrizable completion. As a consequence every (LF)-space is angelic and in this way the answer to a question posed by K. Floret is given. Some contributions to the general problem of regularity in inductive limits are also given. Particularly, we prove for every (LF)-space the equivalence between the conditions of being: (a) sequentially retractive, (b) sequentially compact-regular, (c) compact-regular and (d) precompactly retractive. At the same time we give an approach to the regularity and retractivity problems for the weak topologies. Let us mention here that our results can also be proved for inductive limits of a large class of locally convex spaces, that includes the metric and dual metric spaces, as we have pointed out it in a recent work [Math. Z. 195, 365-381 (1987; Zbl 0604.46011)].

MSC:

46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46M40 Inductive and projective limits in functional analysis
46A50 Compactness in topological linear spaces; angelic spaces, etc.

Citations:

Zbl 0604.46011
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References:

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