On weighted inductive limits of spaces of continuous functions. (English) Zbl 0575.46025

Bonet, José

Math. Z. 192, 9-20 (1986).

The article is concerned with the problem of projective description of weighted inductive limits of spaces of continuous functions. It was stated by K.-D. Bierstedt, R. Meise and W. H. Summers [Trans. Am. Math. Soc. 272, 107-160 (1982)] as follows:
Let E be a locally convex space, let V be a decreasing sequence of strictly positive weights on a Hausdorff completely regular topological space X and let \(\bar V\) be the maximal Nachbin family associated to V. Determine when (a) \(V_ 0C(X,E)=C\bar V_ 0(X,E)\) and (b) \(VC(X,E)=C\bar V(X,E)\) hold algebraically and topologically.
Some open questions are treated. Concerning (b), it is proved that \(C\bar V(X,E)\) is a (DF)-space for every normed space E if X is normal or if every element of V is continuous and \(\bar V\) has a cofinal family of continuous weights, a known result if X is discrete. As regards to (a), it was known that if E is normed, then (W): “\(V_ 0C(X,E)\) is a topological subspace of \(C\bar V_ 0(X,E)\) for every sequence V of continuous weights on every locally compact space X”. It is proved that a reflexive Fréchet space satisfies (W) if and only if it is a quojection in the sense of Bellenot and Dubinsky. The class of locally convex spaces satisfying (W) is studied. In particular, the space of distributions \({\mathcal D}'(\Omega)\) has (W).

MSC:

46E10	Topological linear spaces of continuous, differentiable or analytic functions
46E40	Spaces of vector- and operator-valued functions
46M40	Inductive and projective limits in functional analysis
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.)
46A04	Locally convex Fréchet spaces and (DF)-spaces

Keywords:

projective description of weighted inductive limits of spaces of continuous functions; Nachbin family; (DF)-space; reflexive Fréchet space; quojection

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