Lineability of summing sets of homogeneous polynomials

TL;DR

This paper investigates the lineability and spaceability properties of the set of points where a continuous homogeneous polynomial between Banach spaces is summing, contributing to the understanding of the structure of summing sets.

Contribution

It introduces new results on the lineability and spaceability of summing sets of homogeneous polynomials in Banach spaces.

Findings

Identification of conditions for lineability of summing sets

Establishment of spaceability results for summing sets

Extension of summing set properties to various Banach space contexts

Abstract

Given a continuous $n$-homogeneous polynomial $P\colon E\longrightarrow F$ between Banach spaces and $1\leq q\leq p<\infty$, in this paper we investigate some properties concerning lineability and spaceability of the $(p;q)$-summing set of $P$, defined by $S_{p;q}(P)=\{a\in E:P\mathrm{is}% (p;q)\mathrm{summing at}a\}$.