Theorems of Krein Milman type for certain convex sets of functions operators
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(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).

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     title = {Theorems of {Krein} {Milman} type for certain convex sets of functions operators},
     journal = {Annales de l'Institut Fourier},
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Phelps, Robert R. Theorems of Krein Milman type for certain convex sets of functions operators. Annales de l'Institut Fourier, Tome 20 (1970) no. 2, pp. 45-54. doi : 10.5802/aif.351. https://www.numdam.org/articles/10.5802/aif.351/

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[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).

  • Nathanson, Ekaterina S.; Jørgensen, Palle E. T. Trotter’s limit formula for the Schrödinger equation with singular potential, Journal of Mathematical Physics, Volume 58 (2017) no. 12 | DOI:10.1063/1.5013243
  • Levashov, V. A. Operator analogs of the Krein - Milman theorem, Functional Analysis and Its Applications, Volume 14 (1980) no. 2, p. 130 | DOI:10.1007/bf01086562
  • Shashkin, Yu. A. Convex sets, extreme points, and simplexes, Journal of Soviet Mathematics, Volume 4 (1975) no. 6, p. 625 | DOI:10.1007/bf01083882

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