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Transactions of the American Mathematical Society

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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
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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\vert \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}\vert L = {\delta _y}\vert L$ or there is $ t \in R,t \ne 0$, such that $ 0 \ne {\delta _x}\vert L = t{\delta _y}\vert 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.

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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