Weak topology

In mathematics, the weak topology on a set, with respect to a collection of functions from that set into topological spaces, is the weakest (that is, smallest) topology on the set which makes all the functions continuous. For example, the product topology is defined to be the weak topology with respect to the projection maps of the product.

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

Φ_x : X ' → R or C

defined by

Φ_x(φ) = φ(x)

remains continuous.

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.