Non-Equivalent Norms on C^b(K)

Let A be a non-zero normed vector space and let K = B (0) 1 be the closed unit ball of A. Also, let φ be a non-zero element of A ∗ such that ∥φ∥ ≤ 1. We first define a new norm ∥ · ∥φ on C b (K), that is a non-complete, non-algebraic norm and also non equivalent to the norm ∥ · ∥∞. We next show that for 0 ̸= ψ ∈ A ∗ with ∥ψ∥ ≤ 1, the two norms ∥ · ∥φ and ∥ · ∥ψ are equivalent if and only if φ and ψ are linearly dependent. Also by applying the norm ∥ · ∥φ and a new product “ · ” on C b (K), we present the normed algebra ( C bφ(K), ∥ · ∥φ ) . Finally we investigate some relations between strongly zero-product preserving maps on C b (K) and C bφ(K).