Sets of weak sequential continuity for polynomials
Let \(P({}^NE)\) be the space of all \(N\)-homogeneous continuous polynomials on the Banach space \(E\). The subspace of all polynomials which are weakly sequentially continuous on \(E\) is denoted by \(P_{wsc}({}^NE)\). The authors deal with properties of the set \(C_P = \{x \in E : P\) is weakly sequentially continuous at \( x \}\), where \(P \in P({}^NE)\). In the first section several basic properties of \(C_P\) are given, the formula \[ C_{P \cdot Q} = (C_P \cap C_Q) \cup (C_P \cap P^{-1}(0)) \cup (C_Q \cap Q^{-1}(0)) \] for all \(P \in P({}^NE)\) and \(Q \in P({}^ME)\) is proved, and partial answers to the following questions are given. Question 1. Given \(P \in P({}^NE)\), does there always exist \(Q \in P({}^{N+1}E)\) such that \(C_P=C_Q\)? Question 2. Given \(P,Q \in P({}^NE)\), does there exist \(R \in P({}^{M}E)\) for some \(M\) such that \(C_R=C_P \cap C_Q\)? Question 3. Given \(P,Q \in P({}^NE)\), does there exist \(R \in P({}^{M}E)\) for some \(M\) such that \(C_R=C_P \cup C_Q\)? In the second section the authors study the set \(C_P\) for separable Banach spaces and Banach spaces with unconditional finite dimensional decompositions (FDD). In this cases the results are sharper and the answers to the above questions are more complete.
weakly sequentially continuous polynomials
unconditional finite dimensional decomposition (FDD)
scientific article; zbMATH DE number 1893647