Abstract: Let X be a Banach lattice, and denote by X_+ its positive cone. The weak topology on X_+ is metrizable if and only if it coincides with the strong topology if and only if X is Banach-lattice isomorphic to l^1(\Gamma) for a set \Gamma. The weak^* topology on X_+^* is metrizable if and only if X is Banach-lattice isomorphic to a C(K)-space, where K is a metrizable compact space.
Keywords: normed lattice; Banach lattice; positive cone; AM-space; AL-space; Banach lattice C(K); Banach lattice l^1(\Gamma); strong topology; weak topology; weak^* topology; coincidence of topologies; metrizability; nonatomic measure
DOI: DOI 10.14712/1213-7243.2024.004
AMS Subject Classification: 46B42 46E05 54E35