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標題: Almost convergence of sequences in Banach spaces in weak, strong, and absolute senses
作者: Li, Y.C.
關鍵字: Banach limit;sigma-limit;alpha sigma-limit;weak almost-convergence;strong almost-convergence;absolute almost-convergence;invariant means
Project: Taiwanese Journal of Mathematics
期刊/報告no:: Taiwanese Journal of Mathematics, Volume 10, Issue 1, Page(s) 209-218.
We introduce concepts of sigma-lim sup and sigma-lim inf for bounded sequences of real numbers and show a Cauchy criterion for sequences of vectors which converge in the sense of a sigma-limit (i.e., absolute almost convergence). Then a sufficient condition on a bounded sequence {{x(n)((m))}(n=1)(infinity) subset of l(infinity) (X) is given for the following equality to hold: a sigma- lim(m ->infinity) sigma-lim(n ->infinity) x(n)((m)) = sigma- lim(n ->infinity) a sigma- lim(m ->infinity) x(n)((m)). Finally, applying this result we show that sigma- lim(n ->infinity) f(sin(n theta)) and sigma- lim(n ->infinity) f (cos(n theta)) exist whenever f is a weakly continuous function on [-1, 1] with values in a reflexive Banach space.
ISSN: 1027-5487
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