Theorems of Krein Milman type for certain convex sets of functions operators
Phelps, Robert R.
Annales de l'Institut Fourier, Tome 20 (1970), p. 45-54 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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(X).

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(X).

@article{AIF_1970__20_2_45_0,
     author = {Phelps, Robert R.},
     title = {Theorems of Krein Milman type for certain convex sets of functions operators},
     journal = {Annales de l'Institut Fourier},
     volume = {20},
     year = {1970},
     pages = {45-54},
     doi = {10.5802/aif.351},
     mrnumber = {44 \#4501},
     zbl = {0195.40807},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1970__20_2_45_0}
}

Phelps, Robert R. Theorems of Krein Milman type for certain convex sets of functions operators. Annales de l'Institut Fourier, Tome 20 (1970) pp. 45-54. doi : 10.5802/aif.351. http://gdmltest.u-ga.fr/item/AIF_1970__20_2_45_0/

[1] Erret Bishop and R. R. Phelps, The support functionals of a convex set, Proc. Symp. Pure Math. vol 7 (Convexity), A.M.S. (1963), p. 27-35. | MR 27 #4051 | Zbl 0149.08601

[2] R. M. Blumenthal, J. Lindenstrauss and R. R. Phelps, Extreme operators into C(K), Pacific J. Math. 15 (1965), p. 747-756. | MR 35 #758 | Zbl 0141.32101

[3] N. Bourbaki, Espaces vectoriels topologiques, Ch. 1 et 2, 2e édition, Paris, 1966.

[4] N. Dinculeanu, Vector measures, Berlin, 1967.

[5] N. Dunford and J. T. Schwartz, Linear operators Part I, (1958), Interscience. | Zbl 0084.10402

[6] P. D. Morris and R. R. Phelps, Theorems of Krein-Milman type for certain convex sets of operators, Trans. Amer. Math. Soc. 150 (1970), 183-200. | MR 41 #7409 | Zbl 0198.46601

[7] G. Seever, Generalization of a theorem of Lindenstrauss (dittoed notes).