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Regularity of general weighted inductive limits. (English) Zbl 1023.46038

The author considers weighted inductive limits \({\mathcal V}A(X,E)\) of spaces of continuous or measurable functions with values in a normed space \(E\) in a very general setting, cf. K.-D. Bierstedt, R. Meise and W. H. Summers [Trans. Am. Math. Soc. 272, 107-160 (1982; Zbl 0599.46026)] and J. Wengenroth [Results Math. 32, 169-178 (1997; Zbl 0898.46005)]. If \({\mathcal V}\) satisfies (a slight generalization of) Vogt’s condition \((WQ)\), he proves that the inductive limit equals its projective hull \(A\overline V(X,E)\) algebraically, that the two spaces have the same bounded sets, and that the inductive limit is regular. In the scalar \((LF)\)-case, this result is due to K.-D. Bierstedt and J. Bonet [Math. Nachr. 165, 25-48 (1994; Zbl 0839.46015)] resp. to the author [loc. cit.], but the present proof for the more general theorem is simpler than for the special cases; it becomes clear that the reason behind this is that the algebraic projective description always holds in the scalar \((LB)\)-case. The article also contains two completeness results for \({\mathcal V}A(X,E)\) when \(E\) is a Banach space and \(A(X,E)\) is the space of all continuous or measurable \(E\)-valued functions.

MSC:

46E40 Spaces of vector- and operator-valued functions
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46M40 Inductive and projective limits in functional analysis
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