Pe{\l}czy\'{n}ski's type sets and Pe{\l}czy\'{n}ski's geometrical properties of locally convex spaces

TL;DR

This paper introduces new geometric and topological properties of locally convex spaces based on Pe{{}zy's sets, exploring their implications and applications to classical function spaces.

Contribution

It defines and studies new Pe{{}zy's properties of order (p,q) for locally convex spaces, expanding the understanding of their geometric and topological structure.

Findings

Introduction of $(V^*)$ and $(V)$ subsets of order $(p,q)$

Development of Pe{{}zy's properties $V^*$, $V$, and $(u)$ of order $p$

Applications to classical function spaces

Abstract

For $1\leq p\leq q\leq\infty$ and a locally convex space $E$, we introduce and study the $(V^\ast)$ subsets of order $(p,q)$ of $E$ and the $(V)$ subsets of order $(p,q)$ of the topological dual $E'$ of $E$. Using these sets we define and study the (sequential) Pe{\l}czy\'{n}ski's property $V^\ast$ of order $(p,q)$, the (sequential) Pe{\l}czy\'{n}ski's property $V$ of order $(p,q)$, and the Pe{\l}czy\'{n}ski's property $(u)$ of order $p$ in the class of all locally convex spaces. To this end, we also introduce and study several new completeness type properties, weak barrelledness conditions, Schur type properties, the Gantmacher property for locally convex spaces, and $(q,p)$-summing operators between locally convex spaces. Applications to some classical function spaces are given.