Weak topology

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The weak topology on the dual of a Banach space is the smallest topology where the sets {x: a<x(v)<b} are open for v in the Banach space and a, b be real numbers. The weak* topology is the smallest topology on a Banach space where {v: a<x(v)<b} are open for x in the dual of the Banach space and a, b be real numbers. These topologies are the weakest where the application function from B X B* -> R is continuous.