One may call subsets of a topological vector space weakly closed (respectively, compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, differentiable, analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak 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 fewest open sets) such that all elements of X* remain continuous. Explicitly, 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 R or C. 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 being an intersection of finitely many sets of the form φ-1(U).
More generally, if X is a vector space and F is any family of linear functionals on X (in the algebraic dual space), then the initial topology of X with respect to the family F, denoted by σ(X,F), is sometimes also called the weak topology with respect to F. If F=X* is the continuous dual space of X, then the weak topology with respect to F coincides with the weak topology defined above. The weak topology σ(X,F) is induced by the family of seminorms,
In this formulation, 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 .
The weak topology is characterized by the following condition: a net (xλ) in X converges in the weak topology to the element x of X if and only if φ(xλ) converges to φ(x) in R or C for all φ in X* .
In particular, if xn is a sequence in X, then xn converges weakly to x if
as n → ∞ for all φ ∈ X*. In this case, it is customary to write
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*, if X is normed, is itself a normed vector space by using the norm ||φ|| = sup||x||≤1|φ(x)|. This norm gives rise to a topology, called the strong topology, on X*. This is the topology of uniform convergence. The uniform and strong topologies are generally different for other spaces of linear maps; see below.
A space X can be embedded into X** by
for all x in X. In particular, a sequence of φn ∈ X* converges to φ provided that
for all x in X. In this case, one writes
as n → ∞.
Weak-* convergence is sometimes called the topology of simple convergence or the topology of pointwise convergence. Indeed, it coincides with the topology of pointwise convergence of linear functionals.
By definition, the weak-* topology is weaker than the weak topology on X*. An important fact about the weak-* topology is the Banach-Alaoglu theorem: if X is normed, then the unit ball in X* is weak*-compact (more generally, the polar in X* of a neighborhood of 0 in X is weak*-compact). Moreover, the unit ball in a normed space X is compact in the weak topology if and only if X is reflexive.
If a normed space X is separable, then the weak-* topology is metrizable on (norm-)bounded subsets of X*.
as n→∞. Here the notion of convergence corresponds to the norm on L2.
In contrast weak convergence only demands that
for all functions f∈L2 (or, more typically, all f in a dense subset of L2 such as a space of test functions). For given test functions, the relevant notion of convergence only corresponds to the topology used in C.
For example, in the Hilbert space L2(0,π), the sequence of functions
For example, the strong operator topology on L(X,Y) is the topology of pointwise convergence. For instance, if Y is a normed space, then this topology is defined by the seminorms indexed by x∈X:
More generally, if a family of seminorms Q defines the topology on Y, then the seminorms pq,x on L(X,Y) defining the strong topology are given by
indexed by q∈Q and x∈X.