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2006 ALMOST CONVERGENCE OF SEQUENCES IN BANACH SPACES IN WEAK, STRONG, AND ABSOLUTE SENSES
Yuan-Chuan Li
Taiwanese J. Math. 10(1): 209-218 (2006). DOI: 10.11650/twjm/1500403812

Abstract

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 $\{ \{ x^{(m)}_n \}^{\infty}_{n=1} \}^{\infty}_{m=1} \subset \ell^{\infty}(X)$ is given for the following equality to hold: \[ a\sigma - \lim_{m \to \infty} \sigma - \lim_{n \to \infty} x_{n}^{(m)} = \sigma - \lim_{n \to \infty} a\sigma - \lim_{m \to \infty} x_{n}^{(m)}. \] Finally, applying this result we show that $\sigma - \lim\limits_{n \to \infty} f(\sin(n\theta))$ and $\sigma - \lim\limits_{n \to \infty} f(\cos(n\theta))$ exist whenever $f$ is a weakly continuous function on $[−1,1]$ with values in a reflexive Banach space.

Citation

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Yuan-Chuan Li. "ALMOST CONVERGENCE OF SEQUENCES IN BANACH SPACES IN WEAK, STRONG, AND ABSOLUTE SENSES." Taiwanese J. Math. 10 (1) 209 - 218, 2006. https://doi.org/10.11650/twjm/1500403812

Information

Published: 2006
First available in Project Euclid: 18 July 2017

zbMATH: 1161.40300
MathSciNet: MR2186175
Digital Object Identifier: 10.11650/twjm/1500403812

Subjects:
Primary: 46B99, 46C05

Rights: Copyright © 2006 The Mathematical Society of the Republic of China

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