The Convex Sets in Banach Spaces and Polynomial Approximation

A Banach space A, an open subset V of A, and an open subset U of A' are considered. Our definition introduces novel categories of topological algebras of holomorphic functions on A. We demonstrate the equality of the two sets of holomorphic functions (Hwv(V)) and (Hw*vk(U)) under specific assumptions. We demonstrated that norm-dense Pgi(A) is found in Pw(A) and norm-dense Pgi*(A') is found in Pw*(A'). Additionally, we demonstrated that Pgi(A) is τk-dense in Hwvk(V) and Pgi*(A') is τk*-dense in Hw*vk(U) for a Banach space with a decreasing Schauder basis A, a polynomially convex weakly open subset V of A, and a polynomially convex weak-star open subset U of A.