topology, then becomes a Hausdorff complete locally convex topological vector space (TVS). For any subset let If are any subsets then and where denotes the convex balanced hull of For any finite-dimensional vector subspace of let denote the Euclidean topology on which is the unique topology that makes into a Hausdorff topological vector space (TVS). If denotes the union of all closures as varies over all finite dimensional vector subspaces of then (see this footnote for an explanation). If is an absorbing subset of then by the Banach–Alaoglu theorem, is a weak-* compact subset of If is any non-empty subset of a vector space and if is any vector space of linear functionals on (that is, a vector subspace of the algebraic dual space of ) then the real-valued map defined by is a seminorm on If then by definition of the supremum, so that the map defined above would not be real-valued and consequently, it would not be a seminorm. Continuous dual space Suppose that is a topological vector space (TVS) with continuous dual space The important special case where and the brackets represent the canonical map: is now considered. The triple is the called the associated with The polar of a subset with respect to this canonical pairing is: For any subset where denotes the closure of in The Banach–Alaoglu theorem states that if is a neighborhood of the origin in then and this polar set is a compact subset of the continuous dual space when is endowed with the weak-* topology (also known as the topology of pointwise convergence). If satisfies for all scalars of unit length then one may replace the absolute value signs by (the real part operator) so that: The prepolar of a subset of is: If satisfies for all scalars of unit length then one may replace the absolute value signs with so that: where The bipolar theorem characterizes the bipolar of a subset of a topological vector space. If is a normed space and is the open or closed unit ball in (or even any subset of the closed unit ball that contains the open unit ball) then is the closed unit ball in the continuous dual space when is endowed with its canonical dual norm. Geometric definition for cones The polar cone of a convex cone is the set This definition gives a duality on points and hyperplanes, writing the latter