Scholarlink

<p>In this paper, we introduce a notion of some class of subsets (good positioned sets) of a topological vector space and prove some corespondences between closures of algebraic sum of sets and sum of their closures in suitable topologies (Theorem 2.1). Using this idea, we also prove Theorem 2.5 from which as a corollary we obtain a well known Dieudonne theorem concerning the closedness of algebraic sum of sets. We also obtain some properties regarding the closedness of the algebraic sum of subsets of reflexive Banach spaces and subsets of finite-dimensional spaces (Corollary 2.3 and Corollary 2.4). We give also a nontrivial example of situation where Diudonne theorem cannot be used to determine a closedness of the algebraic sum, but we conclude it from Theorem 2.1 (Example 2.7).</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Some remarks on closedness of algebraic sum of sets

  • 期刊论文

  • 发表日期：

    2026年1月14日

查看全文
  • Hubert Przybycień

摘要

In this paper, we introduce a notion of some class of subsets (good positioned sets) of a topological vector space and prove some corespondences between closures of algebraic sum of sets and sum of their closures in suitable topologies (Theorem 2.1). Using this idea, we also prove Theorem 2.5 from which as a corollary we obtain a well known Dieudonne theorem concerning the closedness of algebraic sum of sets. We also obtain some properties regarding the closedness of the algebraic sum of subsets of reflexive Banach spaces and subsets of finite-dimensional spaces (Corollary 2.3 and Corollary 2.4). We give also a nontrivial example of situation where Diudonne theorem cannot be used to determine a closedness of the algebraic sum, but we conclude it from Theorem 2.1 (Example 2.7).