Convergent subsequences from sequences of functions

Author:
James L. Thornburg

Journal:
Trans. Amer. Math. Soc. **208** (1975), 171-192

MSC:
Primary 40A05; Secondary 26A45, 46A40

DOI:
https://doi.org/10.1090/S0002-9947-1975-0372458-8

MathSciNet review:
0372458

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a sequence of functions, where is a nonempty subset of the -dimensional Euclidean space and is an ordered vector space with positive cone . If , sufficient conditions are given that have a subsequence such that for each the sequence is monotone for sufficiently large, depending on . If each is an ordered topological vector space, sufficient conditions are given that has a subsequence such that for every the sequence is either monotone for sufficiently large depending on , or else the sequence is convergent. If for each and a Banach space then a definition of bounded variation is given so that if is uniformly norm bounded and the variation of the functions is uniformly bounded then there is a convergent subsequence of . In the case for each and is such that bounded monotone sequences converge then the given conditions imply the existence of a subsequence of which converges for each .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0372458-8

Keywords:
Sequences of functions,
convergence of subsequences,
monotonicity of subsequences,
variation,
functions of bounded variation,
ordered topological vector spaces

Article copyright:
© Copyright 1975
American Mathematical Society