Convergent subsequences from sequences of functions
Author:
James L. Thornburg
Journal:
Trans. Amer. Math. Soc. 208 (1975), 171192
MSC:
Primary 40A05; Secondary 26A45, 46A40
MathSciNet review:
0372458
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Abstract 
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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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 P. Hartman, Ordinary differential equations, Wiley, New York, 1964. MR 30 #1270. MR 0171038 (30:1270)
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 E. W. Hobson, The theory of functions of a real variable and the theory of Fourier's series, Dover, New York, 1927.
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 L. K. Jackson and K. Schrader, Existence and uniqueness of solutions of boundary value problems for third order differential equations, J. Differential Equations 9 (1971), 4654. MR 42 #4813. MR 0269920 (42:4813)
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 W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. Vol. 1, NorthHolland, Amsterdam, 1971.
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 K. Schrader and J. Thornburg, Sufficient conditions for the existence of convergent subsequences, Pacific J. Math. (to appear). MR 0374860 (51:11056)
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 H. H. Schaefer, Topological vector spaces, Macmillan, New York, 1966. MR 33 #1689. MR 0193469 (33:1689)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197503724588
PII:
S 00029947(1975)03724588
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
