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(۰\right)}_۱} (the closed unit ball of A). Also for a non-zero element \varphi \in A^* with \left\| \varphi \right\| \leq ۱ , 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.

کلیدواژه ها:

Best approximation ، E-proximinal ، Quasi Chebyshev ، Strongly proximinal