Weak Topology - The Weak and Strong Topologies

The Weak and Strong Topologies

Let K be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications K will be either the field of complex numbers or the field of real numbers with the familiar topologies. Let X be a topological vector space over K. Namely, X is a K vector space equipped with a topology so that vector addition and scalar multiplication are continuous.

We may define a possibly different topology on X using the continuous (or topological) dual space X*. The topological dual space consists of all linear functions from X into the base field K which are continuous with respect to the given topology. The weak topology on X is the initial topology with respect to X*. In other words, it is the coarsest topology (the topology with the fewest open sets) such that each element of X* is a continuous function. In order to distinguish the weak topology from the original topology on X, the original topology is often called the strong topology.

A subbase for the weak topology is the collection of sets of the form φ-1(U) where φ ∈ X* and U is an open subset of the base field K. 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, 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 (Rudin 1991, Theorem 3.10).