Sets of Weak Sequential Continuity for Polynomials

Let P : E → K be an N−homogeneous polynomial, where E is a Banach space over K = R or C . We study properties of the set CP = {x ∈ E : P is weakly sequentially continuous at x}. Introduction Our interest in set of points of weak sequential continuity of a polynomial arises from the following simple observations. If P is any 2−homogeneous scalar valued polynomial on E which is weakly sequentially continuous at 0, then P is weakly sequentially continuous at every point of E. However, the analogous result for 3−homogeneous polynomials is false. (We shall recall the simple details for these observations, as well as the necessary background material, below.) Given an N−homogeneous polynomial P : E → K , we let CP = {x ∈ E : P is weakly sequentially continuous at x}. Our aim in this paper is to study CP . In Section 1, we examine general properties of this set, obtaining for example a formula for CP ·Q. This formula will enable us to obtain information about non-reducibility of polynomials, and our techniques will also yield information about, for example, 3 and 4−homogeneous polynomials on l2. Later in this section, we raise and given partial answers to the following questions: (a). Given P ∈ P(E), does there exist Q ∈ P(E) such that CP = CQ? (b). Given P and Q ∈ P(E), does there exist a polynomial R such that CR = CP ∩CQ, or such that CR = CP ∪ CQ? In Section 2, we focus our attention on properties of CP when the underlying space E is separable, or has an unconditional finite dimensional decomposition. Our methods shed light on the structure of certain spaces of polynomials and, at several places in the text we have inserted examples to illustrate this. Our examples will be restricted to lp−spaces; note that for spaces with the Dunford Pettis property, every polynomial is weakly sequentially continuous, and so there are no examples of interest for these spaces. The same occurs with T ′ (the dual of Tsirelson’s original space), and in fact there are Banach spaces E without the Dunford-Pettis property such that both P(E) = Pwsc( E) and P(E ) = Pwsc( E ) for every n ∈ N ([C-G-G], Theorem 5.4). 1991 Mathematics Subject Classification. Primary 46G25; Secondary 47H60, 46B25, 46B45.