Complemented subspaces of homogeneous polynomials

TL;DR

This paper investigates the conditions under which certain subspaces of homogeneous polynomials are complemented, showing that containing an isomorphic copy of c0 prevents complementarity.

Contribution

It establishes that subspaces of compact and weakly continuous homogeneous polynomials containing c0 are not complemented in the full polynomial space.

Findings

Subspace of compact polynomials containing c0 is not complemented.

Subspace of weakly continuous polynomials containing c0 is not complemented.

Provides conditions for non-complementarity in polynomial spaces.

Abstract

Let $\mathcal{P}_{K} (^{n}E; F)$ (resp. $\mathcal{P}_{w} (^{n}E; F)$) the subspace of all $P\in \mathcal{P}(^{n}E; F)$ which are compact (resp. weakly continuous on bounded sets). We show that if $\mathcal{P}_{K} (^{n}E; F)$ contains an isomorphic copy of $c_{0}$, then $\mathcal{P}_{K} (^{n}E; F)$ is not complemented in $\mathcal{P}(^{n}E; F)$. Likewise we show that if $\mathcal{P}_{w} (^{n}E; F)$ contains an isomorphic copy of $c_{0}$, then $\mathcal{P}_{w}(^{n}E; F)$ is not complemented in $\mathcal{P}(^{n}E; F)$.