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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
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
Summary:
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.
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