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This chapter introduces (after Cascales and Orihuela) a large class of locally convex spaces under the name of the class \(\mathfrak {G}\) . The class \(\mathfrak {G}\) contains among others all \((LM)\) -spaces (hence \((LF)\) -spaces), and dual metric spaces (hence \((DF)\) -spaces), spaces of distributions \(D^{\prime }(\varOmega )\) , and spaces \(A(\varOmega )\) of real analytic functions on open \(\varOmega \subset \mathbb {R}^n\) . We show (following Cascales and Orihuela) that every precompact set in a lcs in the class \(\mathfrak {G}\) is metrizable. This general result covers many already known theorems for \((DF)\) -spaces, \((LF)\) -spaces, and dual metric spaces.

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Metrizability of Compact Sets in the Class \(\mathfrak {G}\)

  • 章节

  • 发表日期：

    2025年

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  • Jerzy Kąkol,
  • Wiesław Kubiś,
  • Manuel López-Pellicer,
  • Damian Sobota

摘要

This chapter introduces (after Cascales and Orihuela) a large class of locally convex spaces under the name of the class \(\mathfrak {G}\) . The class \(\mathfrak {G}\) contains among others all \((LM)\) -spaces (hence \((LF)\) -spaces), and dual metric spaces (hence \((DF)\) -spaces), spaces of distributions \(D^{\prime }(\varOmega )\) , and spaces \(A(\varOmega )\) of real analytic functions on open \(\varOmega \subset \mathbb {R}^n\) . We show (following Cascales and Orihuela) that every precompact set in a lcs in the class \(\mathfrak {G}\) is metrizable. This general result covers many already known theorems for \((DF)\) -spaces, \((LF)\) -spaces, and dual metric spaces.