Notes on the geometry of space of polynomials

Abstract

We show that the symmetric injective tensor product space ${\hat{\otimes }}_{n,s,\epsilon }E$ is not complex strictly convex if E is a complex Banach space of $\dim E\ge 2$ and if $n\ge 2$ holds. It is also reproved that ${\ell }_{\infty }$ is finitely represented in ${\hat{\otimes }}_{n,s,\epsilon }E$ if E is infinite dimensional and if $n\ge 2$ holds, which was proved in the other way by Dineen.