Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Operator minimax theorems in Banach lattices

Author: Carl G. Looney
Journal: Proc. Amer. Math. Soc. 65 (1977), 303-308
MSC: Primary 49B40; Secondary 47H99
MathSciNet review: 0451113
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \psi :X \times Y \to (E, \leqslant ,\left\Vert \cdot \right\Vert)$, where X and Y are convex, X is compact, and E is a dedekind complete Banach lattice with unit e. If each $ \psi ( \cdot ,y)$ is continuous $ \leqslant $-concave on X, $ \{ \psi ( \cdot ,y):y \in Y\} $ is convex, and $ \psi (X \times Y)$ is minorized in E, then $ \sup \inf \psi (x,y) = \inf \sup \psi (x,y)$. Similar theorems are included.

References [Enhancements On Off] (What's this?)

  • [1] G. Choquet, Lectures on analysis. Vols. I, II, III, Benjamin, New York, 1969.
  • [2] N. Dinculeanu, Vector measures, International Series of Monographs in Pure and Applied Mathematics, Vol. 95, Pergamon Press, Oxford-New York-Toronto, Ont.; VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. MR 0206190
  • [3] Ky Fan, Minimax theorems, Proc. Nat. Acad. Sci. U. S. A. 39 (1953), 42–47. MR 0055678
  • [4] Graham Jameson, Ordered linear spaces, Lecture Notes in Mathematics, Vol. 141, Springer-Verlag, Berlin-New York, 1970. MR 0438077
  • [5] Hellmuth Kneser, Sur un théorème fondamental de la théorie des jeux, C. R. Acad. Sci. Paris 234 (1952), 2418–2420 (French). MR 0050249
  • [6] J. L. Kelley and Isaac Namioka, Linear topological spaces, With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. MR 0166578
  • [7] W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. Vol. I, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1971. North-Holland Mathematical Library. MR 0511676
  • [8] Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470
  • [9] J. von Neuman, Zur theorie der Gellschaftsspiele, Math. Ann. 100 (1928), 295-320.
  • [10] Anthony L. Peressini, Ordered topological vector spaces, Harper & Row, Publishers, New York-London, 1967. MR 0227731

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 49B40, 47H99

Retrieve articles in all journals with MSC: 49B40, 47H99

Additional Information

Keywords: Minimax theorem, Banach lattice, convex operator, convex sets, quasiconvex operators, Dinculeanu representation theorem
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society