A particularly important example of a weak topology is that on a normed vector space with respect to its (continuous) dual. The remainder of this article will deal with this case.
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 (x_{n}) in X converges in the weak topology to the element x of X if and only if φ(x_{n}) converges to φ(x) for all φ in X '.
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.
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 BanachAlaoglu 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.
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