Geometry of spaces of real polynomials of degree at most $n$

Abstract

We study the geometry of the unit ball of the space of integral polynomials of degree at most $n$ on a real Banach space. We prove Smul'yan type theorems for Gâteaux and Fréchet differentiability of the norm on preduals of spaces of polynomials of degree at most $n$. We show that the set of extreme points of the unit ball of the predual of the space of integral polynomials is $\{\pm {\sum }_{j=0}^n{\varphi }^j:\varphi \in E^{\prime },\Vert \varphi \Vert \le 1\}$. This contrasts greatly with the situation for homogeneous polynomials where the set of extreme points of the unit ball is the set $\{\pm {\varphi }^n:\varphi \in E^{\prime },\Vert \varphi \Vert =1\}$.