Taiwanese Journal of Mathematics

ON $\sigma$-LIMIT AND $s\sigma$-LIMIT IN BANACH SPACES

Yuan-Chuan Li

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Abstract

For bounded sequences in a normed linear space $X$, we introduce a notion of limit, called the $s\sigma$-limit, and discuss some interesting properties related to $\sigma$-limit and $s\sigma$-limit. It is shown that the space $X_{s\sigma}$ (resp. $X_\sigma$) of all $s\sigma$-convergent (resp. $\sigma$-convergent) sequences in $X$ is a Banach space, and the space $\mathbb{C}_{s\sigma}$ is a unital Banach subalgebra of $\ell^\infty$ such that every Banach limit restricted to $\mathbb{C}_{s\sigma}$ is a multiplicative linear functional. We also use $s\sigma$-limit to characterize continuity of functions and prove two versions of the dominated convergence theorem in terms of $\sigma$-limit and $s\sigma$-limit.

Article information

Source
Taiwanese J. Math., Volume 9, Number 3 (2005), 359-371.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500407845

Digital Object Identifier
doi:10.11650/twjm/1500407845

Mathematical Reviews number (MathSciNet)
MR2162882

Zentralblatt MATH identifier
1099.46018

Subjects
Primary: 42A99: None of the above, but in this section 46B25: Classical Banach spaces in the general theory 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12]

Keywords
Banach limit weak almost convergence strong almost convergence $\sigma$-limit $s\sigma$-limit Bochner integrable Lebesgue dominated convergence theorem

Citation

Li, Yuan-Chuan. ON $\sigma$-LIMIT AND $s\sigma$-LIMIT IN BANACH SPACES. Taiwanese J. Math. 9 (2005), no. 3, 359--371. doi:10.11650/twjm/1500407845. https://projecteuclid.org/euclid.twjm/1500407845


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