as the intersection of two oppositely-oriented half-spaces. The polar hyperplane of a point is the locus ; the dual relationship for a hyperplane yields that hyperplane's polar point. Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article. Properties Unless stated otherwise, will be a pairing. The topology is the weak-* topology on while is the weak topology on For any set denotes the real polar of and denotes the absolute polar of The term "polar" will refer to the polar. The (absolute) polar of a set is convex and balanced. The real polar of a subset of is convex but necessarily balanced; will be balanced if is balanced. If for all scalars of unit length then is closed in under the weak-*-topology on . A subset of is weakly bounded (i.e. -bounded) if and only if is absorbing in . For a dual pair where is a TVS and is its continuous dual space, if is bounded then is absorbing in If is locally convex and is absorbing in then is bounded in Moreover, a subset of is weakly bounded if and only if is absorbing in The bipolar of a set is the -closed convex hull of that is the smallest -closed and convex set containing both and Similarly, the bidual cone of a cone is the -closed conic hull of If is a base at the origin for a TVS then If is a locally convex TVS then the polars (taken with respect to ) of any 0-neighborhood base forms a fundamental family of equicontinuous subsets of (i.e. given any bounded subset of there exists a neighborhood of the origin in such that ). Conversely, if is a locally convex TVS then the polars (taken with respect to ) of any fundamental family of equicontinuous subsets of form a neighborhood base of the origin in Let be a TVS with a topology Then is a locally convex TVS topology if and only if is the topology of uniform convergence on the equicontinuous subsets of The last two results explain why equicontinuous subsets of the continuous dual space play such a prominent role in the modern theory of functional analysis: because equicontinuous subsets encapsulate all information about the locally convex space 's original topology. Set relations and For all scalars and for all real and However, for the real