Local properties of polynomials on a Banach space (Q5954252)

If \(E\) is a complex Banach space then a map \(P: E \mapsto \mathbb C\) is called a continuous \(n\)-homogeneous polynomial if there is a continuous \(n\)-linear map \(L: E^n \mapsto \mathbb C\) such that \(P(x) = L(x,x,\ldots,x)\) for all \(x \in E\). The space of all such polynomials with \(\|P\|:= \sup_{\|x\|\leq 1} |P(x)|\) is denoted by \(\mathcal{P}\)\((^n E)\). Clearly, \(\mathcal{P}\)\((^1 E)\) is the usual dual space. By analogy with the definition of a smooth point, a unit vector \(x_0\) in \(E\) is called a smooth point of order \(n\) of the unit ball if there is a unique element \(P \in \mathcal{P}\)\((^n E)\) with \(1 = \|x_0\|= P(x_0) = \|P\|\). This notion is essentially complex since for real spaces there are no smooth points of order \(2\) (or higher). The authors characterize the set of all smooth points of order \(n\) for the complex spaces \(c_0, \ell_p, L_p\) and \(C(K)\). In the last section the dual notion of locally uniformly rotund points of \(\mathcal{P}\)\((^n E)\) is investigated.