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Pełczyński's type sets and Pełczyński's geometrical properties of locally convex spaces

Saak Gabriyelyan

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) Pełczyński's property $V^\ast$ of order $(p,q)$, the (sequential) Pełczyński's property $V$ of order $(p,q)$, and the Pełczyński'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.

Pełczyński's type sets and Pełczyński's geometrical properties of locally convex spaces

Abstract

For and a locally convex space , we introduce and study the subsets of order of and the subsets of order of the topological dual of . Using these sets we define and study the (sequential) Pełczyński's property of order , the (sequential) Pełczyński's property of order , and the Pełczyński's property of order 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 -summing operators between locally convex spaces. Applications to some classical function spaces are given.
Paper Structure (18 sections, 164 theorems, 217 equations)

This paper contains 18 sections, 164 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Completeness type properties in locally convex spaces
  4. Convergent and summable sequences in topological vector spaces
  5. $p$-(quasi)barrelled locally convex spaces
  6. $p$-Schur property for locally convex spaces
  7. $(V^\ast)$ type subsets of locally convex spaces
  8. $(V)$ type subsets of locally convex spaces
  9. $V$ types of weak barrelledness conditions
  10. $V$ type properties for locally convex spaces
  11. $V^\ast$ type properties for locally convex spaces
  12. Property $p$-$(u)$ for locally convex spaces
  13. Weakly sequentially $p$-(pre)compact and $p$-convergent operators
  14. Pełczyński's type sets and $p$-convergent operators
  15. Gantmacher's property for locally convex spaces and an operator characterization of Pełczyński's property $sV_p$
  16. ...and 3 more sections

Key Result

Theorem 1.2

For every compact space $K$, the Banach space $C(K)$ has the property $V$.

Theorems & Definitions (372)

  • Definition 1.1: Pelcz-62
  • Theorem 1.2: Pelcz-62
  • Definition 1.3: Pelcz-58
  • Definition 1.4
  • Definition 1.5: LCCD
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 362 more