Every normed vector space X is, by using the norm to measure distances, a metric space and hence a topological space. This topology on X is also called the strong topology. The weak topology on X is defined using the continuous dual space X '. This dual space consists of all linear functions from X into the base field R or C which are continuous with respect to the strong topology. The weak topology on X is the weakest topology (the topology with the least open sets) such that all elements of X ' remain continuous. Explicitly, 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 being an intersection of finitely many sets of the form φ-1(U) with φ in X ' and U an open subset of the base field R or C. A sequence (xn) in X converges in the weak topology to the element x of X if and only if φ(xn) converges to φ(x) for all φ in X '.
The dual space X ' is itself a normed vector space by using the norm ||φ|| = sup||x||≤1|φ(x)|. This norm gives rise to the strong topology on X '. One may also define a weak* topology on X ' by requiring that it be the weakest topology such that for every x in X, the substitution map
An important fact about the weak* topology is the Banach-Alaoglu theorem: the unit ball in X ' is compact in the weak* topology.
Furthermore, the unit ball of X is compact in the weak topology if and only if X is reflexive.