Theorems of Krein Milman type for certain convex sets of functions operators
Annales de l'Institut Fourier, Volume 20 (1970) no. 2, pp. 45-54.

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).

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).

@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},
     pages = {45--54},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {20},
     number = {2},
     year = {1970},
     doi = {10.5802/aif.351},
     mrnumber = {44 #4501},
     zbl = {0195.40807},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.351/}
}
TY  - JOUR
AU  - Phelps, Robert R.
TI  - Theorems of Krein Milman type for certain convex sets of functions operators
JO  - Annales de l'Institut Fourier
PY  - 1970
SP  - 45
EP  - 54
VL  - 20
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.351/
DO  - 10.5802/aif.351
LA  - en
ID  - AIF_1970__20_2_45_0
ER  - 
%0 Journal Article
%A Phelps, Robert R.
%T Theorems of Krein Milman type for certain convex sets of functions operators
%J Annales de l'Institut Fourier
%D 1970
%P 45-54
%V 20
%N 2
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.351/
%R 10.5802/aif.351
%G en
%F 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, Volume 20 (1970) no. 2, pp. 45-54. doi : 10.5802/aif.351. http://www.numdam.org/articles/10.5802/aif.351/

[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 | Zbl

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

[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

[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 | Zbl

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

Cited by Sources: