We study this concept and investigate its relationship with the other convergences, and also we introduce the $mn$-topology on normed Riesz algebras.
📌 3 Key Takeaways
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Objective & Hypothesis:A net $(x_\alpha)_{\alpha\in A}$ in an $f$-algebra $E$ is called multiplicative order convergent to $x\in E$ if $\lvert x_\alpha-x\rvert\cdot u \rightarrow 0$ for all $u\in E_+$.
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Key Findings & Data:A net $(x_\alpha)_{\alpha\in A}$ in a normed Riesz algebra $E$ is said to be multiplicative norm convergent to $x\in E$ if $\big\lVert \lvert x_\alpha-x\rvert\cdot u\big\rVert\to 0$ for each $u\in E_+$.
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Conclusion & Impact:We study this concept and investigate its relationship with the other convergences, and also we introduce the $mn$-topology on normed Riesz algebras.
A net $(x_\alpha)_{\alpha\in A}$ in an $f$-algebra $E$ is called multiplicative order convergent to $x\in E$ if $\lvert x_\alpha-x\rvert\cdot u \rightarrow 0$ for all $u\in E_+$. This convergence was introduced and studied on $f$-algebras with the order convergence. In this paper, we study a variation of this convergence for normed Riesz algebras with respect to the norm convergence. A net $(x_\alpha)_{\alpha\in A}$ in a normed Riesz algebra $E$ is said to be multiplicative norm convergent to $x\in E$ if $\big\lVert \lvert x_\alpha-x\rvert\cdot u\big\rVert\to 0$ for each $u\in E_+$. We study this concept and investigate its relationship with the other convergences, and also we introduce the $mn$-topology on normed Riesz algebras.
Aydın, A. (2022). The multiplicative norm convergence in normed Riesz algebras. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 1, 24–32. https://ajindex.com/articles/529515
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