, denoted by or simply , making all maps continuous, as ranges over . The weak topology on is now automatically defined as described in the article Dual system. However, for clarity, we now repeat it. Definition. The weak topology on induced by (and ) is the weakest topology on , denoted by or simply , making all maps continuous, as ranges over . If the field has an absolute value , then the weak topology on is induced by the family of seminorms, , defined by for all and . This shows that weak topologies are locally convex. Assumption. We will henceforth assume that is either the real numbers or the complex numbers . Canonical duality We now consider the special case where is a vector subspace of the algebraic dual space of (i.e. a vector space of linear functionals on ). There is a pairing, denoted by or , called the canonical pairing whose bilinear map is the canonical evaluation map, defined by for all and . Note in particular that is just another way of denoting i.e. . Assumption. If is a vector subspace of the algebraic dual space of then we will assume that they are associated with the canonical pairing . In this case, the weak topology on (resp. the weak topology on ), denoted by (resp. by ) is the weak topology on (resp. on ) with respect to the canonical pairing . The topology is the initial topology of with respect to . If is a vector space of linear functionals on , then the continuous dual of with respect to the topology is precisely equal to . The weak and weak* topologies Let be a topological vector space (TVS) over , that is, is a vector space equipped with a topology so that vector addition and scalar multiplication are continuous. We call the topology that starts with the original, starting, or given topology (the reader is cautioned against using the terms "initial topology" and "strong topology" to refer to the original topology since these already have well-known meanings, so using them may cause confusion). We may define a possibly different topology on using the topological or continuous dual space , which consists of all linear functionals from into the base field that are continuous with respect to the given topology. Recall that is the canonical evaluation map defined by for all and