Weak topology (polar topology)

In functional analysis and related areas of mathematics the weak topology is the coarsest polar topology, the topology with the fewest open sets, on a dual pair. The finest polar topology is called strong topology. In functional analysis and related areas of mathematics the weak topology is the coarsest polar topology, the topology with the fewest open sets, on a dual pair. The finest polar topology is called strong topology. Under the weak topology the bounded sets coincide with the relatively compact sets which leads to the important Bourbaki–Alaoglu theorem. Given a dual pair ( X , Y , ⟨ , ⟩ ) {displaystyle (X,Y,langle , angle )} the weak topology σ ( X , Y ) {displaystyle sigma (X,Y)} is the weakest polar topology on X {displaystyle X} so that That is the continuous dual of ( X , σ ( X , Y ) ) {displaystyle (X,sigma (X,Y))} is equal to Y {displaystyle Y} up to isomorphism.