mean "real polar" (rather than "absolute polar", as is done in this article) and use the notation for it (rather than the notation that is used in this article and in [Narici 2011]). The real bipolar of a subset of sometimes denoted by is the set ; it is equal to the -closure of the convex hull of For a subset of is convex, -closed, and contains In general, it is possible that but equality will hold if is balanced. Furthermore, where denotes the balanced hull of Competing definitions The definition of the "polar" of a set is not universally agreed upon. Although this article defined "polar" to mean "absolute polar", some authors define "polar" to mean "real polar" and other authors use still other definitions. No matter how an author defines "polar", the notation almost always represents choice of the definition (so the meaning of the notation may vary from source to source). In particular, the polar of is sometimes defined as: where the notation is standard notation. We now briefly discuss how these various definitions relate to one another and when they are equivalent. It is always the case that and if is real-valued (or equivalently, if and are vector spaces over ) then If is symmetric (i.e. or equivalently, ) then where if in addition is real-valued then If and are vector spaces over (so that is complex-valued) and if (where note that this implies and ), then where if in addition for all real then Thus for all of these definitions of the polar set of to agree, it suffices that for all scalars of unit length (where this is equivalent to for all unit length scalar ). In particular, all definitions of the polar of agree when is a balanced set (which is often, but not always, the case) so that often, which of these competing definitions is used is immaterial. However, these difference in the definitions of the "polar" of a set do sometimes introduce subtle or important technical differences when is not necessarily balanced. Specialization for the canonical duality Algebraic dual space If is any vector space then let denote the algebraic dual space of which is the set of all linear functionals on The vector space is always a closed subset of the space of all -valued functions on under the topology of pointwise convergence so when is endowed with the subspace