Theorems of Krein Milman type for certain convex sets of functions operators

Phelps, Robert R.

Annales de l'Institut Fourier, Tome 20 (1970) no. 2, pp. 45-54.

On donne des conditions suffisantes sous lesquelles, pour un convexe borné fermé $B$ d’un espace localement convexe réel $E$, l’ensemble $C(X,B)$ [des fonctions continues de l’espace compact $X$ dans $B$] est l’enveloppe convexe uniformément fermée dans $C(X,E)$ de ses points extrémaux. On applique ces résultats à la boule unité de l’espace d’opérateurs bornés (ou compacts, ou faiblement compacts) de certains espaces de Banach dans $C\left(X\right)$.

Sufficient conditions are given in order that, for a bounded closed convex subset $B$ of a locally convex space $E$, the set $C(X,B)$ of continuous functions from the compact space $X$ into $B$, is the uniformly closed convex hull in $C(X,E)$ of its extreme points. Applications are made to the unit ball of bounded (or compact, or weakly compact) operators from certain Banach spaces into $C\left(X\right)$.

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