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

Commentationes mathematicae Universitatis Carolinae

article

Zbigniew Lipecki

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;   

DOI  :  10.14712/1213-7243.2022.006

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 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.

【 授权许可】

CC BY