Grothendieck locally convex spaces of continuous vector valued functions

Abstract

Let ^{X, E) be the space of continuous functions from the completely regular Hausdorff space X into the Hausdorff locally convex space E, endowed with the compact-open topology. Our aim is to characterize the ^(X, E) spaces which have the following property: weak-star and weak sequential convergences coincide in the equicontinuous subsets of ^(X, E)'. These spaces are here called Grothendieck spaces. It is shown that in the equicontinuous subsets of E' the σ(E', E)-and β(E', ^-sequential convergences coincide, if ^(X, E) is a Grothendieck space and X contains an infinite compact subset. Conversely, if X is a G-space and E is a strict inductive limit of Frechet-Montel spaces ^(X, E) is a Grothendieck space. Therefore, it is proved that if £ is a separable Frechet space, then E is a Montel space if and only if there is an infinite compact Hausdorff X such that , E) is a Grothendieck space.1. Introduction. In this paper X will always denote a completely regular Hausdorff topological space, E a Hausdorff locally convex space, and &( X, E) the space of continuous functions from Xinto E, endowed with the compact-open topology. When E is the scalar field of reals or complex numbers, we write ^(X) instead ^( X, E).It is well known that Ή{X, E) is a Montel space whenever ^(X) and E so are, hence, if and only if X is discrete and E is a Montel space (see [5] , [16]).We study what happens when X has the following weaker property: the compact subsets of X are G-spaces (see below for definitions).We obtain in Theorem 4.4 that if £ is a Frechet-Montel space and X has that property, then ^( X, E) is a Grothendieck locally convex space. The key in the proof is the following fact: every countable equicontinuous subset of ^(X, E)' lies, via a Radon-Nikodym theorem, in a suitable L\τ 9 Eβ). As a consequence of a theorem of Mύjica [10], the same result is true when E is a strict inductive limit of Frechet-Montel spaces.In §3 we study the converse of 4.4. In Corollary 3.3 it is proved that if X contains an infinite compact subset, E is a Frechet separable space and #( X, E) is a Grothendieck space, then E is a Montel space. This property characterizes the Montel spaces among the Frechet separable spaces.

Grothendieck Spaces and Duals of Injective Tensor Products

“…The above result of Freniche [13], combined with the fact that C(A)-spaces always have the property (V) of Pelczyriski [18], also gives a positive answer to the conjecture raised by Drewnowski [10], that C{K) contains a complemented copy of c 0 if the space C(K, E) contains a complemented copy of c 0 for some Frechet-Montel space E. Further, by a method that also can be used to study stability of some other properties with respect to injective tensor products, we show that if E, Fare Frechet spaces such that F is Montel and has the approximation property, then E® e F contains a complemented copy of c 0 if and only if E contains a complemented copy of c 0 .…”

“…In this paper we study stability of Grothendieck spaces and of the property of containing a complemented copy of c 0 with respect to injective tensor products of two Frechet spaces when one space is Montel. In [13] Freniche proved that if E is a Frechet-Montel space and A" is a compact Hausdorff space such that C{K) is a Grothendieck space, then C(K, E) is a Grothendieck space. We shall extend this result to injective tensor products E0 e F when E is a Frechet-Montel space, F is a Grothendieck Frechet space and either E or (F b )' b has the approximation property.…”

“…By M(E, F) we denote the space of all Montel maps from E into F, that is, all continuous linear maps from E into F that transform bounded sets into relatively compact sets. If E is a Banach space, then the space of Montel operators coincides with the space of all compact operators from E into F. In [13] Freniche extended the definition of Grothendieck spaces to lcs by defining an lcs E to be Grothendieck if a{E',E)-and o{E\ (.E^-sequential convergence coincide on equicontinuous subsets of E'. For Frechet spaces this definition of Grothendieck spaces coincides with the usual one, since every weak*-converging sequence in the dual is equicontinuous.…”

Weak compactness of certain sets of measures

“…These results are extended to the case of C(X, E) by many authors ( [4], [1], [17]). In this paper we give necessary and sufficient conditions for an equicontinuous subset of F to be σ(F , F )-compact (i.e.…”

Grothendieck operators on tensor products

“…In case of E = C (K) it follows immediately from results of Freniche [Fr2] (compare [DL,Cor. 3.7]) that C (K, F ) [Jh]).…”