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Algebraic genericity of certain families of nets in Functional Analysis

Sheldon Dantas, Daniel L. Rodríguez-Vidanes

Abstract

In Functional Analysis, certain conclusions apply to sequences, but they cannot be carried over when we consider nets. In fact, some nets, including sequences, can behave unexpectedly. In this paper we are interested in exploring the prevalence of these unusual nets in terms of linearity. Each problem is approached with different methods, which have their own interest. As our results are presented in the contexts of topological vector spaces and normed spaces, they generalize or improve a few ones in the literature. We study lineability properties of families of (1) nets that are weakly convergent and unbounded, (2) nets that fail the Banach-Steinhaus theorem, (3) nets indexed by a regular cardinal $κ$ that are weakly dense and norm-unbounded, and finally (4) convergent series which have associated nets that are divergent.

Algebraic genericity of certain families of nets in Functional Analysis

Abstract

In Functional Analysis, certain conclusions apply to sequences, but they cannot be carried over when we consider nets. In fact, some nets, including sequences, can behave unexpectedly. In this paper we are interested in exploring the prevalence of these unusual nets in terms of linearity. Each problem is approached with different methods, which have their own interest. As our results are presented in the contexts of topological vector spaces and normed spaces, they generalize or improve a few ones in the literature. We study lineability properties of families of (1) nets that are weakly convergent and unbounded, (2) nets that fail the Banach-Steinhaus theorem, (3) nets indexed by a regular cardinal that are weakly dense and norm-unbounded, and finally (4) convergent series which have associated nets that are divergent.
Paper Structure (8 sections, 7 theorems, 22 equations)

This paper contains 8 sections, 7 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Our results
  4. Main Results
  5. Weakly convergent and unbounded nets
  6. Nets that fail the Banach-Steinhaus theorem
  7. Weakly dense and norm-unbounded nets
  8. Convergent series with associated divergent nets

Key Result

Theorem 2.1

Let $\aleph_0 \leqslant \kappa \leqslant \mathfrak c$ be a cardinal number. Let $X$ be a real or complex infinite-dimensional TVS. There exists a directed set $\mathcal{A}$ of cardinality $\kappa$ such that the family of nets in $X$ indexed by $\mathcal{A}$ that are unbounded and weakly convergent i

Theorems & Definitions (15)

  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4: Fichtenholz-Kantorovich-Hausdorff theorem
  • Remark 2.5
  • proof : Proof of Theorem \ref{['thm:unbweakly']}
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • Definition 2.8
  • ...and 5 more