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Countability Properties of Weakly Compact Sets in Asymmetric Locally Convex Spaces

Jobst Ziebell

Abstract

A dual pair formulation for asymmetric locally convex spaces is developed that strictly generalises the ordinary vector space setting. The concept of a polar topology carries over to the asymmetric case and some familiar results are reproduced or generalised such as the bipolar theorem and a Mackey-Arens type theorem. The implications of weak compactness and countability properties are studied and appear intimately connected to separation properties. An asymmetric analogue $C_a(K)$ of the well-known space $C_p(K)$ is introduced and the properties of (relatively) (countably respectively sequentially) compact subspaces are investigated. In particular, it is shown that $C_a([0,1])$ is not angelic. However, for Hausdorff subspaces satisfying a simple closure condition, the different compactness conditions are equivalent and imply the Fréchet-Urysohn property. Moreover, an analogue of the Eberlein-Šmulian theorem holds in asymmetrically normed spaces.

Countability Properties of Weakly Compact Sets in Asymmetric Locally Convex Spaces

Abstract

A dual pair formulation for asymmetric locally convex spaces is developed that strictly generalises the ordinary vector space setting. The concept of a polar topology carries over to the asymmetric case and some familiar results are reproduced or generalised such as the bipolar theorem and a Mackey-Arens type theorem. The implications of weak compactness and countability properties are studied and appear intimately connected to separation properties. An asymmetric analogue of the well-known space is introduced and the properties of (relatively) (countably respectively sequentially) compact subspaces are investigated. In particular, it is shown that is not angelic. However, for Hausdorff subspaces satisfying a simple closure condition, the different compactness conditions are equivalent and imply the Fréchet-Urysohn property. Moreover, an analogue of the Eberlein-Šmulian theorem holds in asymmetrically normed spaces.
Paper Structure (8 sections, 47 theorems, 78 equations)

This paper contains 8 sections, 47 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some Topological Notions
  4. Asymmetric Locally Convex Spaces
  5. Asymmetric Dual Pairs
  6. Countability Conditions and Weak Compactness
  7. The space Cₐ(K)
  8. The Weak Topology on an Asymmetrically Normed Space

Key Result

Corollary 2.1

$A$ is relatively compact in $X$ if and only if every net $(x_\alpha)_{\alpha \in I}$ in $A$ has a cluster point in $X$. $A$ is relatively countably compact in $X$ if and only if every sequence $(x_n)_{n \in \mathbb{N}}$ in $A$ has a cluster point in $X$.

Theorems & Definitions (87)

  • Corollary 2.1
  • Definition 2.2: src:KönigKuhn:AngelicSpaces
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 77 more