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 $\big\{\pm\sum_{j=0}^n\phi^j:\phi \in E',\|\phi\|\le 1\big\}$. This contrasts greatly with the situation for homogeneous polynomials where the set of extreme points of the unit ball is the set $\{\pm\phi^n:\phi\in E',\|\phi\|=1\}$.