In other words, a subset of "X" 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 φ−1("U"). More generally, if "F" is a subset of the algebraic dual space, then the initial topology of "X" with respect to "F", denoted by σ("X","F"), is the weak topology with respect to "F" . If one takes "F" to be the whole continuous dual space of "X", then the weak topology with respect to "F" coincides with the weak topology defined above. If the field "K" has an absolute value formula_1, then the weak topology σ("X","F") is induced by the family of  seminorms,  for all "f"∈"F" and "x"∈"X". In particular, weak topologies are locally convex. From this point of view, the weak topology is the coarsest polar topology; see weak topology (polar topology) for details. Specifically, if "F" is a vector space of linear functionals on "X" which separates points of "X", then the continuous dual of "X" with respect to the topology σ("X","F") is precisely equal to "F" . The weak topology is characterized by the following condition: a net ("x""λ") in "X" converges in the weak topology to the element "x" of "X" if and only if φ("x""λ") converges to φ("x") in R or C for all φ in "X*" . In particular, if "x"n is a sequence in "X", then "x"n converges weakly to "x" if  as formula_4 for all formula_5. In this case, it is customary to write  or, sometimes,  If "X" is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and "X" is a locally convex topological vector space. If "X" is a normed space, then the dual space "X*" is itself a normed vector space by using the norm ǁφǁ = supǁxǁ≤1|φ("x")|.