definition of "the weak topology defined on by "). For instance, after defining " distinguishes points of " (resp, " is a total subset of ") as above, then the dual definition of " distinguishes points of " (resp, " is a total subset of ") is immediately obtained. For instance, once is defined then it should be automatically assume that has been defined without mentioning the analogous definition. The same applies to many theorems. Convention: Adhering to common practice, unless clarity is needed, whenever a definition (or result) is given for a pairing then mention the corresponding dual definition (or result) will be omitted but it may nevertheless be used. In particular, although this article will only define the general notion of polar topologies on with being a collection of -bounded subsets of this article will nevertheless use the dual definition for polar topologies on with being a collection of -bounded subsets of Identification of with Although it is technically incorrect and an abuse of notation, the following convention is nearly ubiquitous: Convention: This article will use the common practice of treating a pairing interchangeably with and also denoting by Polar topologies Throughout, is a pairing of vector spaces over the field and is a non-empty collection of -bounded subsets of For every and is convex and balanced and because is a -bounded, the set is absorbing in The polar topology on determined (or generated) by (and ), also called the -topology on or the topology of uniform convergence on the sets of is the unique topological vector space (TVS) topology on for which forms a neighbourhood subbasis at the origin. When is endowed with this -topology then it is denoted by If is a sequence of positive numbers converging to then the defining neighborhood subbasis at may be replaced with without changing the resulting topology. When is a directed set with respect to subset inclusion (i.e. if for all there exists some such that ) then the defining neighborhood subbasis at the origin actually forms a neighborhood basis at Seminorms defining the polar topology Every determines a seminorm defined by where and is in fact the Minkowski functional of Because of this, the -topology on is always a locally convex topology. Modifying If every positive scalar multiple of a set in is contained in some set belonging to then the defining neighborhood subbasis at the origin can be