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.
Citation
Yuan-Chuan Li. "ON $\sigma$-LIMIT AND $s\sigma$-LIMIT IN BANACH SPACES." Taiwanese J. Math. 9 (3) 359 - 371, 2005. https://doi.org/10.11650/twjm/1500407845
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