The Convex Sets in Banach Spaces and Polynomial Approximation

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DOI: https://doi.org/10.28924/2291-8639-22-2024-123
Published: Jul 29, 2024
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Ashraf S. ELshreif, Habeeb Ibrahim, Mohammed E. Dafaalla, Osman Abdalla Adam Osman, The Convex Sets in Banach Spaces and Polynomial Approximation, Int. J. Anal. Appl., 22 (2024), 123.

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Ashraf S. ELshreif, Habeeb Ibrahim, Mohammed E. Dafaalla, Osman Abdalla Adam Osman

Abstract

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.

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