Order intervals in C(K). Compactness, coincidence of topologies, metrizability

[摘要] Let K be a compact space and let C(K) be the Banach lattice of real-valued continuous functions on K. We establish eleven conditions equivalent to the strong compactness of the order interval [0,x] in C(K), including the following ones: (i) 0\} consists of isolated points of K; (ii) [0,x] is pointwise compact; (iii) [0,x] is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on [0,x]; (v) the strong and weak topologies coincide on [0,x]. \noindent Moreover, the weak topology and that of pointwise convergence coincide on [0,x] if and only if 0\} is scattered. Finally, the weak topology on [0,x] is metrizable if and only if the topology of pointwise convergence on [0,x] is such if and only if 0\} is countable.

[学科分类] 物理化学和理论化学

[关键词] real linear lattice;order interval;locally solid;Banach lattice C(K);strongly compact;weakly compact;pointwise compact;coincidence of topologies;metrizable;scattered;Čech-Stone compactification