Continuous functions on totally ordered spaces that are compact in their order topologies (Q1840770)

Below \(L\) denotes a totally ordered space that is compact in its order topology. Two samples of the results obtained in this paper. Theorem A. \(C(L)\) has an equivalent Kadec norm, that is, a norm for which the weak and norm topologies coincide on the unit sphere. Theorem B. The following statements are equivalent: (1) there is an equivalent locally uniformly convex norm on \(C(L)\), which is lower-semicontinuous for the pointwise topology; (2) there is an equivalent strictly convex norm on \(C(L)\); (3) there is a set \(\Gamma\) and a bounded linear injection of \(C(L)\) into \(c_0(\Gamma)\), which is continuous for the topologies of pointwise convergence on \(L\) and \(\Gamma\), respectively. For more details the reader is referred to this interesting paper.