zbMATH — the first resource for mathematics

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).

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
Full Text: DOI EuDML
[1] Bellenot, S.F., Dubinsky, E.: Fréchet spaces with nuclear Köthe quotients. Trans. Am. Math. Soc.273, 579-594 (1982) · Zbl 0494.46001
[2] Bierstedt, K.D.: Tensor products of weighted spaces. Bonner Math. Schriften81, 26-58 (1975) · Zbl 0333.46024
[3] Bierstedt, K.D., Meise, R., Summers, W.H.: A projective description of weighted inductive limits. Trans. Am. Math. Soc.272, 107-160 (1982) · Zbl 0599.46026
[4] Bierstedt, K.D., Meise, R.G., Summers, W.H.: Köthe sets and Köthe sequence spaces, p. 27-91 in Functional Analysis, Holomorphy and Approximation Theory, North-Holland Math, Studies71. Amsterdam-New York-Oxford: North-Holland 1982 · Zbl 0504.46007
[5] Bierstedt, K.D., Meise, R.: Distinguished echelon spaces and the projective description of weighted inductive limits of typeV C(X). To appear in Aspects of Mathematics and its Applications, Elsevier Science Publ. B.V., Amsterdam, New York and Oxford, 1986 · Zbl 0645.46027
[6] Bonet, J.: A projective description of weighted inductive limits of spaces of vector valued continuous functions. Collect. Math.34, 117-124 (1983)
[7] Bonet, J.: The countable neighbourhood property and tensor products. Proc. Edinburgh Math. Soc.28, 207-215 (1985) · Zbl 0563.46001
[8] Bonet, J.: Quojections and projective tensor products. Archiv Math.45, 169-173 (1985) · Zbl 0557.46003
[9] Dierolf, S., Zarnadze, D.N.: A note on strictly regular Fréchet spaces. Archiv Math.42, 549-556 (1984) · Zbl 0525.46004
[10] Floret, K.: Some aspects of the theory of locally convex inductive limits. Functional Analysis: Surveys and Recent Results II. Bierstedt, K.D., Fuchssteiner, B., (eds.) North-Holland Math. Studies38, 1980, p. 205-237
[11] Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Am. Math. Soc.16 (1955)
[12] Hollstein, R.: Inductive limits and ?-tensor products. J. Reine Angew. Math.319, 38-62 (1980) · Zbl 0426.46053
[13] Jarchow, H.: Locally convex spaces. Stuttgart: Teubner 1981 · Zbl 0466.46001
[14] Köthe, G.: Topological vector spaces I and II. Berlin, Heidelberg, New York: Springer 1969 and 1979 · Zbl 0179.17001
[15] Nachbin, L.: Elements of approximation theory. Math. Studies, no 14. Princeton: Van Nostrand 1967 · Zbl 0173.41403
[16] Nachbin, L.: A glance at holomorphic factorization and uniform holomorphy. To appear in Complex Analysis, Functional Analysis and Approximation Theory. Jorge Mujica (ed.). Preprint (November 1984)
[17] Valdivia, M.: Topics in Locally convex spaces. Math. Studies67. Amsterdam: North-Holland 1982
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.