# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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by Jo ao B. Prolla and Silvio Machado
Trans. Amer. Math. Soc. 186 (1973), 247-258 Request permission

## 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.
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