Weak topology

An example of a weak topology that is particularly important in functional analysis 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^* (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 fewest 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^* .

The weak* topology

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
:
defined by
:
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.

See also: