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

### Christopher Boyd

University College Dublin, Ireland### Anthony Brown

University College Dublin, Ireland

## 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 $\{\pm\phi^n:\phi\in E',\|\phi\|=1\}$.

## Cite this article

Christopher Boyd, Anthony Brown, Geometry of spaces of real polynomials of degree at most $n$. Rev. Mat. Iberoam. 33 (2017), no. 4, pp. 1149–1171

DOI 10.4171/RMI/966