Polars, Bipolar Theorem, Polar Topologies

Abstract

In a dual pair 〈E, F〉 one wants to define topologies on E associated with collections of suitable subsets of F. (This generalises the definition of the norm topology on the dual E′ of a Banach space E, in this case for the dual pair 〈E′, E〉.) Such a collection $\mathcal M$ defines a ‘polar topology’ on E, where the corresponding neighbourhoods of zero in E are polars of the members of $\mathcal M$. Examples of such topologies are the weak topology and the strong topology. In the first part of the chapter we define polars and investigate some of their properties.