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

For a non-zero normed linear space A, we consider Cb (K), the complex-valued, bounded and continuous functions space on K with ∥·∥∞, where K = B(0) 1 (the closed unit ball of A). Also for a non-zero element φ ∈ A∗ with ∥φ∥ ≤ 1, we consider the space Cbφ (K) as the linear space Cb (K) with the new norm ∥f∥ φ = ∥fφ∥∞ for all f ∈ Cb (K). Some basic properties such as, proximinality, E-proximinality, strongly proximinality and quasi Chebyshev for certain subsets of Cb (K) are characterized with the norms ∥·∥ φ and ∥·∥∞. Some examples for illustration and for comparison between the norms ∥·∥ φ and ∥·∥∞ on Cb (K) are presented.