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In this chapter, we discuss selected properties of Banach spaces of type \(C(K)\) , where K is a linearly ordered compact space, called a compact line for short. In particular, we present Nakhmanson’s theorem stating that if K is a compact line such that \(C_p(K)\) is a Lindelöf space, then K is second-countable. We also discuss the separable complementation property in the context of compact lines. Compact lines are relatively easy to investigate, yet they form a rich class of spaces and provide several interesting examples. A very special case is the smallest uncountable well-ordered space \(\omega _1\) +1, which appeared several times in the previous chapters. Its space of continuous functions turns out to be a canonical example for several topological and geometric properties of Banach spaces. More complicated compact lines provide examples related to Plichko spaces.

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Spaces of Continuous Functions over Compact Lines

  • 章节

  • 发表日期：

    2025年

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

摘要

In this chapter, we discuss selected properties of Banach spaces of type \(C(K)\) , where K is a linearly ordered compact space, called a compact line for short. In particular, we present Nakhmanson’s theorem stating that if K is a compact line such that \(C_p(K)\) is a Lindelöf space, then K is second-countable. We also discuss the separable complementation property in the context of compact lines. Compact lines are relatively easy to investigate, yet they form a rich class of spaces and provide several interesting examples. A very special case is the smallest uncountable well-ordered space \(\omega _1\) +1, which appeared several times in the previous chapters. Its space of continuous functions turns out to be a canonical example for several topological and geometric properties of Banach spaces. More complicated compact lines provide examples related to Plichko spaces.