Denseness of norm-attaining mappings on Banach spaces
Summary: Let \(X\) and \(Y\) be Banach spaces. Let \(P(^nX \colon Y)\) be the space of all \(Y\)-valued continuous \(n\)-homogeneous polynomials on \(X\). We show that the set of all norm-attaining elements is dense in \(P(^nX \colon Y)\) when the set of u.s.e.\ points of the unit ball \(B_{X}\) is dense in the unit sphere \(S_{X}\). Applying strong peak points instead of u.s.e.\ points, we generalize this result to a closed subspace of \(C_b(M, Y)\), where \(M\) is a complete metric space. For complex Banach spaces \(X\) and \(Y\), let \(A_{b}(B_{X} \colon Y)\) be the Banach space of all bounded continuous \(Y\)-valued mappings \(f\) on \(B_{X}\) whose restrictions \(f|_{B_{X}^{\circ}}\) to the open unit ball are holomorphic. It follows that the set of all norm-attaining elements is dense in \(A_{b}(B_{X} \colon Y)\) if the set of all strong peak points in \(A_{b}(B_{X})\) is a norming subset for \(A_{b}(B_{X})\).
scientific article; zbMATH DE number 5690820
scientific article