Polynomial compactness in Banach spaces (Q1293526)

Let \(X\) be an infinite-dimensional Banach space over \({\mathbb K}\) \(({\mathbb R}\) or \({\mathbb C})\), \({\mathcal P}{(^N}X)\) be the set of all \(N\)-homogeneous continuous polynomials \(P: X\to {\mathbb K}\) where \(N\in {\mathbb N}\), \({\mathcal P}(X)\) be the union of all sets \({\mathcal P}{(^N}X)\) and \(X_{{\mathcal P}(X)}\) be the set \(X\) endowed with the weakest topology making all \(P\in {\mathcal P}(X)\) continuous. A characterization of such Banach spaces \(X\) and \(Y\) are given for which a) \(Z_{{\mathcal P}(Z)}\) (here \(Z = X\times Y\)) is not a topological vector space and b) \(Z_{{\mathcal P}(Z)}\) (here \(Z \simeq X\times Y\)) has a nonlinear topology. The notion of polynomial compactness in Banach spases and the \({\mathcal P}\)-Dunford-Pettis property and the \({\mathcal P}^{\leq N}\)-Dunford-Pettis property are introduced and several equivalent conditions for Banach spaces to have these properties are given. It is shown that the sum of two polynomially compact subsets of Banach space \(X\) is again a polynomially compact set if \(X\) has the \({\mathcal P}\)-Dunford-Pettis property and a characterization for a such real Banach space \(X\) is given for which a bounded subset is relatively compact in \(X_{{\mathcal P}(X)}\) if and only if it is separated from zero by all \(P > 0\) in \({{\mathcal P}(X)}\).