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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 0372458
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Abstract: Let $ \{ {y_k}\} $ be a sequence of functions, $ {y_k} \in {\Pi _{s \in S}}{E_s}$ where $ S$ is a nonempty subset of the $ l$-dimensional Euclidean space and $ {E_s}$ is an ordered vector space with positive cone $ {K_s}$. If $ {y_k} \in {\Pi _{s \in S}}{E_s}$, sufficient conditions are given that $ \{ {y_k}\} $ have a subsequence $ \{ {h_k}\} $ such that for each $ t \in S$ the sequence $ \{ {h_k}(t)\} $ is monotone for $ k$ sufficiently large, depending on $ t$. If each $ {E_s}$ is an ordered topological vector space, sufficient conditions are given that $ \{ {y_k}\} $ has a subsequence $ \{ {h_k}\} $ such that for every $ t \in S$ the sequence $ \{ {h_k}(t)\} $ is either monotone for $ k$ sufficiently large depending on $ t$, or else the sequence $ \{ {h_k}(t)\} $ is convergent. If $ {E_s} = B$ for each $ s$ and $ B$ a Banach space then a definition of bounded variation is given so that if $ \{ {y_k}\} $ is uniformly norm bounded and the variation of the functions $ {y_k}$ is uniformly bounded then there is a convergent subsequence $ \{ {h_k}\}$ of $ \{ {y_k}\}$. In the case $ {E_s} = E$ for each $ s \in S$ and $ E$ is such that bounded monotone sequences converge then the given conditions imply the existence of a subsequence $ \{ {h_k}\} $ of $ \{ {y_k}\} $ which converges for each $ t \in S$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0372458-8
PII: S 0002-9947(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