Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Weighted Grothendieck subspaces

Authors: Jo ao B. Prolla and Silvio Machado
Journal: Trans. Amer. Math. Soc. 186 (1973), 247-258
MSC: Primary 46E10
MathSciNet review: 0402477
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let V be a family of nonnegative upper semicontinuous functions on a completely regular Hausdorff space X. For a locally convex Hausdorff space E, let $C{V_\infty }(X;E)$ be the corresponding Nachbin space, that is, the vector space of all continuous functions f from X into E such that vf vanishes at infinity for all $v \in V$, endowed with the topology given by the seminorms of the type $f| \to \sup \{ v(x)p(f(x));x \in X\}$, where $v \in V$ and p is a continuous seminorm on E. Given a vector subspace L of $C{V_\infty }(X;E)$, the set of all pairs $x,y \in X$ such that either $0 = {\delta _x}|L = {\delta _y}|L$ or there is $t \in R,t \ne 0$, such that $0 \ne {\delta _x}|L = t{\delta _y}|L$, is an equivalence relation, denoted by ${G_L}$, and we define for $(x,y) \in {G_L},g(x,y) = 0$ or t, accordingly. The subsets $K{S_L}$, resp. $W{S_L}$, where $g(x,y) \geq 0$, resp. $g(x,y) \in \{ 0,1\}$, are likewise equivalence relations. The G-hull (resp. KS-hull, WS-hull) of L is the vector subspace $\{ f \in C{V_\infty }(X;E);f(x) = g(x,y)f(y)$ for all $(x,y) \in {G_L}\;({\text {resp}}.\;K{S_L},W{S_L})\}$ and L is a G-space (resp. KS-space, WS-space) if its G-hull (resp. KS-hull, WS-hull) is contained in its closure. This paper is devoted to the characterization, by invariance properties, of the G-spaces resp. KS-spaces and WS-spaces of a given Nachbin space $C{V_\infty }(X;E)$. As an application we derive an infinite-dimensional Weierstrass polynomial approximation theorem, and a Tietze extension theorem for Banach space valued compact mappings.

References [Enhancements On Off] (What's this?)

    K.-D. Bierstedt, Gewichtete Räume Stetiger Vektorwertiger Funktionen und das Injektive Tensorprodukt, Ph. D. Dissertation, Johannes Gutenberg-Universität in Mainz, 1970.
  • Jörg Blatter, Grothendieck spaces in approximation theory, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 120. MR 0493107
  • J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353–367. MR 44116
  • R. Engelking, Outline of general topology, North-Holland Publishing Co., Amsterdam; PWN-Polish Scientific Publishers, Warsaw; Interscience Publishers Division John Wiley & Sons, Inc., New York, 1968. Translated from the Polish by K. Sieklucki. MR 0230273
  • Joram Lindenstrauss and Daniel E. Wulbert, On the classification of the Banach spaces whose duals are $L_{1}$ spaces, J. Functional Analysis 4 (1969), 332–349. MR 0250033, DOI
  • Leopoldo Nachbin, Elements of approximation theory, Van Nostrand Mathematical Studies, No. 14, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0217483
  • Leopoldo Nachbin, Weighted approximation for function algebras and quasi-analytic mappings, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, Ill., 1966, pp. 330–333. MR 0199692
  • Leopoldo Nachbin, João B. Prolla, and Silvio Machado, Concerning weighted approximation, vector fibrations, and algebras of operators, J. Approximation Theory 6 (1972), 80–89. MR 350425, DOI
  • A. Pełczyński, A generalization of Stone’s theorem on approximation, Bull. Acad. Polon. Sci. Cl. III 5 (1957), 105-107. MR 19, 135.
  • M. H. Stone, On the compactification of topological spaces, Ann. Soc. Polon. Math. 21 (1948), 153–160. MR 0026316
  • ---, The generalized Weierstrass approximation theorem. In Buck, R. C., Studies in Modern Analysis, MAA Studies in Mathematics 1 (1962), 30-87.
  • Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46E10

Retrieve articles in all journals with MSC: 46E10

Additional Information

Keywords: Nachbin spaces of continuous vector-valued functions, Grothendieck spaces, Kakutani-Stone spaces, Weierstrass-Stone spaces, polynomial algebras, latticial subspaces, Lindenstrauss-Wulbert subspaces, compact mappings
Article copyright: © Copyright 1973 American Mathematical Society