on e-proximinality and strongly proximinality in complex-valued, bounded and continuous functions space

For a non-zero normed linear space a, we considercb (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 considerthe 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.