## Taiwanese Journal of Mathematics

### ALMOST CONVERGENCE OF SEQUENCES IN BANACH SPACES IN WEAK, STRONG, AND ABSOLUTE SENSES

Yuan-Chuan Li

#### 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.

#### Article information

Source
Taiwanese J. Math., Volume 10, Number 1 (2006), 209-218.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500403812

Digital Object Identifier
doi:10.11650/twjm/1500403812

Mathematical Reviews number (MathSciNet)
MR2186175

Zentralblatt MATH identifier
1161.40300

#### Citation

Li, Yuan-Chuan. ALMOST CONVERGENCE OF SEQUENCES IN BANACH SPACES IN WEAK, STRONG, AND ABSOLUTE SENSES. Taiwanese J. Math. 10 (2006), no. 1, 209--218. doi:10.11650/twjm/1500403812. https://projecteuclid.org/euclid.twjm/1500403812