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

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).

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### References:

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