Resumen de Geometry of spaces of real polynomials of degree at most n

Christopher Boyd, Antony Browne

• 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 {±∑nj=0ϕj:ϕ∈E′,∥ϕ∥≤1}. This contrasts greatly with the situation for homogeneous polynomials where the set of extreme points of the unit ball is the set {±ϕn:ϕ∈E′,∥ϕ∥=1}.