Polynomial sequential continuity on \(C(K,E)\) spaces
Let \(E\) be a Banach space. A sequence \((x^n) \subseteq E\) is said to be \(\tau_{\leq N}\) convergent to \(x \in E\) if \(P(x^n)\) tends to \(P(x)\) for all \(P \in P(^kE)\) and \(k \leq N\). The authors consider properties like the following one. The space \(E\) has property \(P_{N,M}\) if \(T(x_1^n,\dots, x_M^n)\) tends to \(T(x_1,\dots,x_M)\) for all \(T \in L_s(^ME)\) and all sequences \((x_j^n)\) with \(\tau_{\leq N}\)-\(\lim x_j^n=x_j\), \(j=1,\dots,M\), where \(N\in\mathbb{N}\cup\{\infty\}\), \(M\in\mathbb{N}\). In this paper several relations among the properties \(P_{N,M}\), \(SP_{N,M}\), \(Q_{N,M}\), and \(SQ_{N,M}\) and relations to the polynomial Schur property and polynomial Dunford-Pettis property are given. Roughly speaking, the main results show that a space \(E\) has one of the above properties iff every (or some) \(C(K,E)\) has this property too, where \(K\) is a scattered compact Hausdorff space. In this way polynomial sequential convergence is connected with pointwise polynomial sequential convergence but the results are different from those obtained in the case of weak sequential convergence.
scientific article; zbMATH DE number 1956921
scientific article