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Uniformly closed sublattices of finite codimension

Eugene Bilokopytov, Vladimir G. Troitsky

Abstract

The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (or sublattice) of finite codimension may be written as an intersection of uniformly closed subspaces (respectively, sublattices) of codimension one. Every uniformly closed sublattice of codimension $n$ contains a uniformly closed ideal of codimension at most $2n$. If the vector lattice is uniformly complete then every ideal of finite codimension is uniformly closed. Results of the paper extend (and are motivated by) results of [AL90a,AL90b] and , as well as Kakutani's characterization of closed sublattices of $C(K)$ spaces.

Uniformly closed sublattices of finite codimension

Abstract

The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (or sublattice) of finite codimension may be written as an intersection of uniformly closed subspaces (respectively, sublattices) of codimension one. Every uniformly closed sublattice of codimension contains a uniformly closed ideal of codimension at most . If the vector lattice is uniformly complete then every ideal of finite codimension is uniformly closed. Results of the paper extend (and are motivated by) results of [AL90a,AL90b] and , as well as Kakutani's characterization of closed sublattices of spaces.
Paper Structure (6 sections, 41 theorems, 11 equations)

This paper contains 6 sections, 41 theorems, 11 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Closed sublattices of $C(\Omega)$
  3. Closed sublattices of Banach sequence spaces
  4. Uniformly closed subspaces of finite codimension
  5. Uniformly closed sublattices of finite codimension
  6. Finite codimensional ideals

Key Result

Lemma 1.1

Let $F$ be a closed subspace of codimension $n$ in a normed space $E$. Then the closure $\overline{F}$ of $F$ in the completion $\overline{E}$ of $E$ has codimension $n$.

Theorems & Definitions (77)

  • Lemma 1.1
  • proof
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • Remark 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Corollary 2.7
  • ...and 67 more