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On Dugundji's notion of positive definiteness


Author: Chi Song Wong
Journal: Proc. Amer. Math. Soc. 46 (1974), 443-450
MSC: Primary 54H25
DOI: https://doi.org/10.1090/S0002-9939-1974-0418076-9
MathSciNet review: 0418076
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Abstract: Dugundji's notion of positive definiteness is generalized to nonnegative real-valued functions on a uniform space. Its relations with completeness and various notions of compactness are investigated. For an arbitrary uniform space $ X$, there may be lack of the right kind of lower semicontinuous real-valued functions on $ X$ and so a further generalization of Dugundji's notion of positive definiteness is needed for the development of the fixed point (or coincidence) theory. With such an extension, a very general fixed point theorem is obtained to include a recent result of the author, which contains, as special cases, some results of S. Banach, F.E. Browder, D. W. Boyd and J. S. W. Wong, M. Edelstein and R. Kannan.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0418076-9
Keywords: Bolzano-Weierstrass property, compactness, completeness, countable compactness, fixed point, lower semicontinuity, pseudo compactness, positive definiteness, sequential compactness, uniformity, uniform continuity
Article copyright: © Copyright 1974 American Mathematical Society

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