Document Type : Research Paper

Authors

Faculty of Mathematical Sciences, Shahrood University of Technology, P. O. Box 3619995161-316, Shahrood, Iran.

Abstract

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.

Keywords

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