In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing. Preliminaries A pairing is a triple consisting of two vector spaces over a field (either the real numbers or complex numbers) and a bilinear map A dual pair or dual system is a pairing satisfying the following two separation axioms: separates/distinguishes points of : for all non-zero there exists such that and separates/distinguishes points of : for all non-zero there exists such that Polars The polar or absolute polar of a subset is the set Dually, the polar or absolute polar of a subset is denoted by and defined by In this case, the absolute polar of a subset is also called the prepolar of and may be denoted by The polar is a convex balanced set containing the origin. If then the bipolar of denoted by is defined by Similarly, if then the bipolar of is defined to be Weak topologies Suppose that is a pairing of vector spaces over Notation: For all let denote the linear functional on defined by and let Similarly, for all let be defined by and let The weak topology on induced by (and ) is the weakest TVS topology on denoted by or simply making all maps continuous, as ranges over Similarly, there are the dual definition of the weak topology on induced by (and ), which is denoted by or simply : it is the weakest TVS topology on making all maps continuous, as ranges over Weak boundedness and absorbing polars It is because of the following theorem that it is almost always assumed that the family consists of -bounded subsets of Dual definitions and results Every pairing can be associated with a corresponding pairing where by definition There is a repeating theme in duality theory, which is that any definition for a pairing has a corresponding dual definition for the pairing Convention and Definition: Given any definition for a pairing one obtains a dual definition by applying it to the pairing If the definition depends on the order of and (e.g. the definition of "the weak topology defined on by ") then by switching the order of and it is meant that this definition should be applied to (e.g. this gives us the