Notes on the geometry of space of polynomials

TL;DR

This paper investigates the geometric properties of symmetric injective tensor product spaces over complex Banach spaces, showing they lack complex strict convexity under certain conditions and revisiting the finite representability of le_infty.

Contribution

It demonstrates the non-strict convexity of symmetric injective tensor products for complex Banach spaces of dimension at least two and re-establishes the finite representability of le_infty in these spaces.

Findings

Symmetric injective tensor product spaces are not complex strictly convex for dim E  ge 2 and n ge 2.

le_infty is finitely represented in these tensor spaces when E is infinite dimensional.

The results extend understanding of the geometric structure of tensor product spaces in functional analysis.

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.