Geometry of the space of integral polynomials

LASSALLE, S.

Alcoy, España

Congreso; VI Encuentro de Análisis Funcional Murcia-Valencia; 2009

Universidad Politécnica de Valencia - Universidad de Murcia

The geometric study of Banach spaces tells us that there are certainsubsets on the unit sphere of a Banach space which are fundamental in our understanding of the geometry of Banach spaces. Such sets include the extreme points, exposed points, and weak$^*$-exposed and strongly exposed points.We study the geometry of the space of $n$-homogeneous integralpolynomials defined on a real Banach space $E$, $\mathcal P_I(^nE)$. We show that when $E'$ has the approximation property and $\widehat{\bigotimes}_{n,s,\epsilon} E$, the $n$-symetric inyective tensor product of $E$, does not contain copies of $\ell_1$, the set of extreme points of the unit ball of $P_I(^nE)$ is equal to ${\pm\phi^n: \phi \in E', ||phi||=1}$. Under the additional assumption that $E'$ has a countable norming set we see that the set of exposed points of the unit ball of $P_I(^nE)$ is also equal to ${\pm\phi^n: \phi \in E', ||phi||=1}$.The above results apply to all real separable Asplund Banach  spaces $E$ such that $E'$ has the approximation property.