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Limited type subsets of locally convex spaces

Saak Gabriyelyan

Abstract

Let $1\leq p\leq q\leq\infty.$ Being motivated by the classical notions of limited, $p$-limited and coarse $p$-limited subsets of a Banach space, we introduce and study $(p,q)$-limited subsets and their equicontinuous versions and coarse $p$-limited subsets of an arbitrary locally convex space $E$. Operator characterizations of these classes are given. We compare these classes with the classes of bounded, (pre)compact, weakly (pre)compact and relatively weakly sequentially (pre)compact sets. If $E$ is a Banach space, we show that the class of coarse $1$-limited subsets of $E$ coincides with the class of $(1,\infty)$-limited sets, and if $1<p<\infty$, then the class of coarse $p$-limited sets in $E$ coincides with the class of $p$-$(V^\ast)$ sets of Pełczyński. We also generalize a known theorem of Grothendieck.

Limited type subsets of locally convex spaces

Abstract

Let Being motivated by the classical notions of limited, -limited and coarse -limited subsets of a Banach space, we introduce and study -limited subsets and their equicontinuous versions and coarse -limited subsets of an arbitrary locally convex space . Operator characterizations of these classes are given. We compare these classes with the classes of bounded, (pre)compact, weakly (pre)compact and relatively weakly sequentially (pre)compact sets. If is a Banach space, we show that the class of coarse -limited subsets of coincides with the class of -limited sets, and if , then the class of coarse -limited sets in coincides with the class of - sets of Pełczyński. We also generalize a known theorem of Grothendieck.
Paper Structure (5 sections, 41 theorems, 41 equations)

This paper contains 5 sections, 41 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries results
  3. Limited-type sets in locally convex spaces
  4. Coarse $p$-limited sets
  5. Limited type sets and $p$-convergent operators

Key Result

Theorem 1.2

Let $E$ be a Banach space. Then:

Theorems & Definitions (89)

  • Definition 1.1
  • Theorem 1.2: BourDies
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 79 more