On E-Proximinality and Strongly Proximinality in Complex-Valued, Bounded and Continuous Functions Space
Document Type : Research Paper
Rasul Tourani
Ali Reza Khoddami
Faculty of Mathematical Sciences, Shahrood University of Technology, P. O. Box 3619995161-316, Shahrood, Iran.
https://doi.org/10.22130/scma.2022.555067.1124
For a non-zero normed linear space $A$, we consider $ C^b\left(K\right) $, the complex-valued, bounded and continuous functions space on $K$ with $ \left\| \cdot \right\|_\infty $, where $ K = \overline{B^{\left(0\right)}_1} $ (the closed unit ball of $A$). Also for a non-zero element $\varphi \in A^*$ with $ \left\| \varphi \right\| \leq 1 $, we consider the space $ C^{b\varphi}\left(K\right) $ as the linear space $ C^b\left(K\right) $ with the new norm $ \left\| f \right\|_\varphi = \left\| f\varphi \right\|_\infty $ for all $ f \in C^b\left(K\right) $. Some basic properties such as, proximinality, E-proximinality, strongly proximinality and quasi Chebyshev for certain subsets of $ C^b\left(K\right) $ are characterized with the norms $ \left\| \cdot \right\|_\varphi $ and $ \left\|\cdot \right\|_\infty $. Some examples for illustration and for comparison between the norms $ \left\| \cdot \right\|_\varphi $ and $ \left\| \cdot \right\|_\infty $ on $ C^b\left(K\right) $ are presented.
Best approximation
E-proximinal
Quasi Chebyshev
Strongly proximinal
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Tourani, R., & Khoddami, A. R. (2023). On E-Proximinality and Strongly Proximinality in Complex-Valued, Bounded and Continuous Functions Space. Sahand Communications in Mathematical Analysis, 20(1), 119-136. doi: 10.22130/scma.2022.555067.1124
Rasul Tourani; Ali Reza Khoddami. "On E-Proximinality and Strongly Proximinality in Complex-Valued, Bounded and Continuous Functions Space". Sahand Communications in Mathematical Analysis, 20, 1, 2023, 119-136. doi: 10.22130/scma.2022.555067.1124
Tourani, R., Khoddami, A. R. (2023). 'On E-Proximinality and Strongly Proximinality in Complex-Valued, Bounded and Continuous Functions Space', Sahand Communications in Mathematical Analysis, 20(1), pp. 119-136. doi: 10.22130/scma.2022.555067.1124
Tourani, R., Khoddami, A. R. On E-Proximinality and Strongly Proximinality in Complex-Valued, Bounded and Continuous Functions Space. Sahand Communications in Mathematical Analysis, 2023; 20(1): 119-136. doi: 10.22130/scma.2022.555067.1124