Geometry of spaces of real polynomials of degree at most nn

  • 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 nn 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 nn. We show that the set of extreme points of the unit ball of the predual of the space of integral polynomials is {±j=0nϕj:ϕE,ϕ1}\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 {±ϕn:ϕE,ϕ=1}\{\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 nn. Rev. Mat. Iberoam. 33 (2017), no. 4, pp. 1149–1171

DOI 10.4171/RMI/966