[Paper Review] The Schur l1 Theorem for filters

Avilés, Antonio, Bernardo Cascales Salinas+2 more|ArXiv.org|Mar 3, 2009

Approximation Theory and Sequence Spaces8 references21 citations

TL;DR

This paper generalizes the classical Schur ℓ₁ theorem—on the equivalence of weak and norm convergence in ℓ₁—by introducing filter-based convergence. It characterizes filters for which weak ℱ-convergence implies norm convergence (Schur filters), introduces the double Schur property for filters, and establishes that filters with this property are both WC1 and Schur filters. The key contribution is a complete characterization of such filters via trace and product constructions.

ABSTRACT

We study the classes of filters F on N such that the weak and strong F-convergence of sequences in l1 coincide. We study also an analogue of l1 weak sequential completeness theorem for filter convergence.

Motivation & Objective

• To extend the classical Schur ℓ₁ theorem to the setting of filter convergence in ℓ₁ spaces.
• To identify necessary and sufficient conditions for a filter ℱ to satisfy the Schur property: weak ℱ-convergence implies norm convergence in ℓ₁.
• To study the weak sequential completeness property (WC1) in the context of filter convergence.
• To introduce and analyze the double Schur property for filters, defined via the product filter ℱ×ℱ.
• To investigate the structure of filter classes, including quasi-increasing, decreasing, and increasing properties, and their interplay with the Schur and WC1 properties.

Proposed method

• Define ℱ-convergence of sequences in topological spaces using filters on a countable set ℕ, generalizing ordinary convergence.
• Introduce the trace of a filter ℱ on an ℱ-stationary set I, denoted ℱ(I), and use it to relate ℱ-convergence to convergence along subsets.
• Define the product filter ℱ₁×ℱ₂ on N₁×N₂ and the sum filter ℱ₁+ℱ₂ on N₁∪N₂ to analyze joint convergence behavior.
• Introduce the double Schur property: a filter ℱ has this property if ℱ×ℱ is a Schur filter, i.e., weak ℱ×ℱ-convergence in ℓ₁ implies norm convergence.
• Use duality between weak and norm convergence in ℓ₁ and the completeness of ℓ₁ to show that double Schur filters imply WC1 and Schur properties.
• Analyze filter classes via domination and inheritance: define quasi-increasing, decreasing, and increasing properties to study closure under taking traces and superfilters.

Experimental results

Research questions

• RQ1 For which filters ℱ does weak ℱ-convergence of a sequence in ℓ₁ imply norm convergence?
• RQ2 What is the relationship between the double Schur property and the Schur or WC1 properties of a filter?
• RQ3 How do filter properties behave under taking traces ℱ(I) for ℱ-stationary sets I?
• RQ4 Can the class of almost Schur filters be explicitly characterized, and does it form a basis for the class of all such filters?
• RQ5 Are the Schur and double Schur properties preserved under filter domination or inheritance via traces?

Key findings

• A filter ℱ has the Schur property if and only if every trace ℱ(I) for I ∈ ℱ* is an almost Schur filter.
• Every filter with the double Schur property is both a WC1-filter and a Schur filter.
• The double Schur property implies that ℱ×ℱ-convergence of (xₙ − xₘ) to 0 in the weak topology of ℓ₁** implies ℓ₁-norm convergence to 0.
• The class of Schur filters is not decreasing: there exist filters ℱ₁ ∈ Schur and ℱ₂ ∉ Schur such that ℱ₁ is dominated by ℱ₂.
• The class of almost Schur filters is quasi-increasing but not decreasing, and the problem of constructing a basis for this class remains open.
• The double Schur property is strictly stronger than the Schur property, and filters dominated by a double Schur filter are necessarily WC1-filters.