Extreme and exposed points of spaces of integral polynomials

Boyd, Christopher; Lassalle, Silvia

doi:10.1090/S0002-9939-09-10158-2

Extreme and exposed points of spaces of integral polynomials
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by Christopher Boyd and Silvia Lassalle PDF

Proc. Amer. Math. Soc. 138 (2010), 1415-1420 Request permission

Abstract:

We show that if $E$ is a real Banach space such that $E’$ has the approximation property and such that $\ell _1\not \hookrightarrow {\widehat \bigotimes _{n,s,\epsilon }} E$, then the set of extreme points of the unit ball of $\mathcal {P}_I(^nE)$ is equal to $\{\pm \phi ^n\colon \phi \in E’,\|\phi \|=1\}$. Under the additional assumption that $E’$ has a countable norming set, we see that the set of exposed points of the unit ball of $\mathcal {P}_I(^nE)$ is also equal to $\{\pm \phi ^n\colon \phi \in E’,\|\phi \|=1\}$.

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