A class of sets in a Banach space coarser than limited sets

TL;DR

This paper introduces coarse p-limited sets in Banach spaces, a new class based on weak* p-summable sequences, and explores their properties and relationships with compact sets and operators.

Contribution

It defines and analyzes coarse p-limited sets, a novel class in Banach space theory, expanding understanding of set classifications and operator interactions.

Findings

Coarse p-limited sets are distinct from compact and weakly compact sets.

The paper establishes relationships between coarse p-limited sets and bounded linear operators.

Properties of coarse p-limited sets are characterized in various Banach space contexts.

Abstract

A wide new class of subsets of a Banach space $X$ named coarse $p$-limited sets ($ 1\leq p < \infty$) is introduced by considering weak* $p$-summable sequences in $X'$ instead of weak* null sequences. We study its basic properties and compare it with the class of compact and weakly compact sets. Results concerning the relationship of coarse $p$-limited sets with operators are obtained.