Convergent subsequences from sequences of functions

Thornburg, James L.

doi:10.1090/S0002-9947-1975-0372458-8

Convergent subsequences from sequences of functions
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by James L. Thornburg PDF

Trans. Amer. Math. Soc. 208 (1975), 171-192 Request permission

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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