Abstract
Let W be a linear subspace of a weighted space of type CV(X,E) and an A-module (A being a subalgebra of C(X)). The weighted approximation problem, as formulated by L. Nachbin [12] in the case E =\(\mathbb{K}\) and for W⊂CVo(X), asks for a description of the closure of W. In its bounded case (see [12]) a general answer was given by W.H. Summers [20] for W⊂CVo(X) which is, under certain additional assumptions on X and V, also valid for W⊂CVo(X,E) as J.B. Prolla [15] had already proved. By extending the methods developed in [10] it is shown in this paper that these results can be generalized to arbitrary X,V and E, and moreover, that by considering compactifications of X similar solutions can be found for subspaces of CV(X,E). Applying them to a result of K.-D. Bierstedt [2] the approximation property of (certain subspaces of) CV(X) can be derived. As a further application tensor products of weighted spaces are investigated.
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BIERSTEDT, K.-D.: Gewichtete Räume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt. I. J.f.d. reine u. angew. Math.259, 186–210 (1973)
—: Gewichtete Räume stetiger vektorwertiger Funktionen und das injektive Tensorprodukt. II. J.f.d. reine u. angew. Math.260, 133–146 (1973)
—: Injektive Tensorprodukte und Slice-Produkte gewichteter Räume stetiger Funktionen. J.f.d. reine u. angew. Math.266, 121–131 (1974)
-: The approximation property for weighted function spaces. to appear in Bonner Math. Schriften (Proc. Conference Bonn, 1974)
-: Tensor products of weighted spaces, to appear in Bonner Math. Schriften (Proc. Conference Bonn, 1974)
BUCHWALTER, H.: Topologies et compactologies. Publ. Dépt. Math. Lyon6–2, 1–74 (1969)
DINCULEANU, N.: Vector measures. 1. Aufl. Berlin: VEB Deutscher Verlag der Wissenschaften 1967
FUCHSSTEINER, B.: Sandwich theorems and lattice semi-groups. J. Functional Analysis16, 1–14 (1974)
GLICKSBERG, I.: Stone-Čech compactifications of products. Trans. Amer. Math. Soc.90, 369–382 (1959)
KLEINSTÜCK, G.: Duals of weighted spaces of continuous functions. to appear in Bonner Math. Schriften (Proc. Conference Bonn, 1974)
KÖNIG, H.: Sublineare Funktionale. Arch. Math.23, 500–508 (1972)
NACHBIN, L.: Weighted approximation for algebras and modules of continuous functions: real and self-adjoint complex cases. Ann. of Math.81, 289–302 (1965)
—: Elements of approximation theory. First ed. Princeton-Toronto-London-Melbourne: Van Nostrand Company (1967)
PROLLA, J.B.: Weighted spaces of vector-valued continuous functions. Ann. Mat. Pura Appl. (4)89, 145–158 (1971)
—: Bishop's generalized Stone-Weierstrass theorem for weighted spaces. Math. Ann.191, 283–289 (1971)
—: The weighted Dieudonné theorem for density in tensor products. Indag. Math.33, 170–175 (1971)
SUMMERS, W.H.: A representation theorem for biequicontinuous completed tensor products of weighted spaces. Trans. Amer. Math. Soc.146, 121–132 (1969)
—: Dual spaces of weighted spaces. Trans. Amer. Math. Soc.151, 323–333 (1970)
SUMMERS, W.H.: The bounded case of the weighted approximation problem. p. 177–183 in “Functional analysis and applications (Symposium Recife, Brasil 1972)”, Springer Lecture Notes in Math.384 (1974)
-: Weighted approximation for modules of continuous functions II. Proc. Symp. Rio de Janeiro (1972), to appear (Hermann, Paris)
-: to appear (cf. Abstracts of Communications, Intern. Congress of Math., Vancouver, 1974)
BIERSTEDT, K.-D., GRAMSCH, B., MEISE, R.: erscheint noch
NACHBIN, L., MACHADO, S., PROLLA, J.B.: Weighted approximation, vector fibrations and algebras of operators. J. Math. pures et appl.50, 299–323 (1971)
BUCHWALTER, H.: Produit topologique, produit tensoriel et c-repletion. Bull. Soc. math. France, Mém.31–32, 51–71 (1972)
PUPIER, R.: Méthodes fonctorielles en topologie générale. Thèse Fac. Sc. Lyon, 1–121 (1971)
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Kleinstück, G. Der beschränkte Fall des gewichteten Approximationsproblems für vektorwertige Funktionen. Manuscripta Math 17, 123–149 (1975). https://doi.org/10.1007/BF01154086
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DOI: https://doi.org/10.1007/BF01154086