On E-Proximinality and Strongly Proximinality in Complex-Valued, Bounded and Continuous Functions Space

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.