Weighted Grothendieck subspaces

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.