, where in particular, . Definition. The weak topology on is the weak topology on with respect to the canonical pairing . That is, it is the weakest topology on making all maps continuous, as ranges over . Definition: The weak topology on is the weak topology on with respect to the canonical pairing . That is, it is the weakest topology on making all maps continuous, as ranges over . This topology is also called the weak* topology. We give alternative definitions below. Weak topology induced by the continuous dual space Alternatively, the weak topology on a TVS is the initial topology with respect to the family . In other words, it is the coarsest topology on X such that each element of remains a continuous function. A subbase for the weak topology is the collection of sets of the form where and is an open subset of the base field . In other words, a subset of is open in the weak topology if and only if it can be written as a union of (possibly infinitely many) sets, each of which is an intersection of finitely many sets of the form . From this point of view, the weak topology is the coarsest polar topology. Weak convergence The weak topology is characterized by the following condition: a net in converges in the weak topology to the element of if and only if converges to in or for all . In particular, if is a sequence in , then converges weakly to if as for all . In this case, it is customary to write or, sometimes, Other properties If is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and is a locally convex topological vector space. If is a normed space, then the dual space is itself a normed vector space by using the norm This norm gives rise to a topology, called the strong topology, on . This is the topology of uniform convergence. The uniform and strong topologies are generally different for other spaces of linear maps; see below. Weak-* topology The weak* topology is an important example of a polar topology. A space can be embedded into its double dual X** by Thus is an injective linear mapping, though not necessarily surjective (spaces for which this canonical embedding is surjective are called reflexive). The weak-* topology on is the