Spaces of Continuous Functions over Compact Lines
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 ω 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.
KeywordsBanach Space Compact Space Double Arrow Interval Topology Convex Classis
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