Simple Proofs for Bochner-Schoenberg-Eberlein and the Bochner-Schoenberg-Eberlein Module Properties on ℓpX,A.

Let X be a nonempty set, A be a commutative Banach algebra, and 1 ≤ p < ∞. In this paper, we present a concise proof for the result concerning the BSE (Banach space extension) property of ℓ p X , A . Specifically, we establish that ℓ p X , A possesses the BSE property if and only if X is finite and A is BSE. Additionally, we investigate the BSE module property on Banach ℓ p X , A -modules and demonstrate that a Banach space ℓ p X , Y serves as a BSE Banach ℓ p X , A -module if and only if X is finite and Y represents a BSE Banach A -module. [ABSTRACT FROM AUTHOR]