# Revista Matemática Iberoamericana

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**Volume 33, Issue 4, 2017, pp. 1149–1171**

**DOI: 10.4171/RMI/966**

Published online: 2017-11-17

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

Christopher Boyd^{[1]}and Anthony Brown

^{[2]}(1) University College Dublin, Ireland

(2) University College Dublin, Ireland

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

*Keywords: *Polynomials on Banach spaces, Gâteaux and Fréchet differentiability, extreme points

Boyd Christopher, Brown Anthony: Geometry of spaces of real polynomials of degree at most $n$. *Rev. Mat. Iberoamericana* 33 (2017), 1149-1171. doi: 10.4171/RMI/966