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Infinite Doubly Stochastic Matrices

Published online by Cambridge University Press:  20 November 2018

J.R. Isbell
J.R. Isbell*
Affiliation:
University of Washington
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This note proves two propositions on infinite doubly stochastic matrices, both of which already appear in the literature: one with an unnecessarily sophisticated proof (Kendall [2]) and the other with the incorrect assertion that the proof is trivial (Isbell [l]). Both are purely algebraic; so we are, if you like, in the linear space of all real doubly infinite matrices A = (aij).

Proposition 1. Every extreme point of the convex set of ail doubly stochastic matrices is a permutation matrix.

Kendall's proof of this depends on an ingenious choice of a topology and the Krein-Milman theorem for general locally convex spaces [2]. The following proof depends on practically nothing: for example, not on the axiom of choice.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 5 , Issue 1 , January 1962 , pp. 1 - 4
Copyright
Copyright © Canadian Mathematical Society 1962

References

Isbell, J. R., Birkhoff's Problem 111, Proc. Amer. Math. Soc. 6(1955), 217-218.Google Scholar
Kendall, M. G., On infinite doubly stochastic matrices and Birkhoff's Problem 111, J. London Math. Soc. 35(1960), 81-84.Google Scholar
Rattray, B. A. and Peck, J. E. L,., Infinite doubly stochastic matrices, Trans. Roy. Soc. Canada 111(3), 49(1955), 55-57.Google Scholar