Weak topology

In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a normed vector space or topological vector space with respect to its (continuous) dual. The remainder of this article will deal with this case, which is one of the basic concepts of functional analysis.

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 strong and weak topologies

Let X be a topological vector space, then in particular X is a topological space carrying a topology as part of its definition. (For example, a normed vector space X is, by using the norm to measure distances, a metric space and hence also a topological vector space.) This topology is also called the strong topology on X.

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,

for all f∈F and x∈X. In particular, weak topologies are locally convex.

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 (Rudin 1991, Theorem 3.10).

Weak convergence

The weak topology is characterized by the following condition: a net (x_λ) in X converges in the weak topology to the element x of X if and only if φ(x_λ) converges to φ(x) in R or C for all φ in X* .

In particular, if x_n is a sequence in X, then x_n converges weakly to x if

as n → ∞ for all φ ∈ X^*. In this case, it is customary to write

or, sometimes,

Other properties

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.

The weak-* topology

A space X can be embedded into X** by

where

T_x(φ) = φ(x).

Thus T : X → X** is an injective linear mapping, though it is not surjective unless X is reflexive. The weak-* topology on X* is the weak topology induced by the image of T: T(X) ⊂ X**. In other words, it is the coarsest topology such that the maps T_x : X* → F remain continuous.

Weak-* convergence

A net φ_λ in X* is convergent to φ in the weak-* topology if it converges pointwise:

for all x in X. In particular, a sequence of φ_n ∈ 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.

Other properties

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*.

Examples

Hilbert spaces

Consider, for example, the difference between strong and weak convergence of functions in the Hilbert space L²(Rⁿ). Strong convergence of a sequence ψ_n∈L²(Rⁿ) to an element ψ means that

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

form an orthonormal basis. In particular, the (strong) limit of ψ_n as n→∞ does not exist. On the other hand, by the Riemann-Lebesgue lemma, the weak limit exists and is zero.

Distributions

One normally obtains spaces of distributions by forming the strong dual of a space of test functions (such as the compactly supported smooth functions on Rⁿ). In an alternative construction of such spaces, one can take the weak dual of a space of test functions inside a Hilbert space such as L². Thus one is lead to consider the idea of a rigged Hilbert space.

Operator topologies

If X and Y are topological vector spaces, the space L(X,Y) of continuous linear operators f:X → Y may carry a variety of different possible topologies. The naming of such topologies depends on the kind of topology one is using on the target space Y to define operator convergence (Yosida 1980, IV.7 Topologies of linear maps). There are, in general, a vast array of possible operator topologies on L(X,Y), whose naming is not entirely intuitive.

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 p_q,x on L(X,Y) defining the strong topology are given by

indexed by q∈Q and x∈X.

In particular, see the weak operator topology and weak* operator topology.

References

• Conway, John B. (1994), A Course in Functional Analysis (2nd ed.), Springer-Verlag, ISBN 0-387-97245-5 
• Pedersen, Gert (1989), Analysis Now, Springer, ISBN 0-387-96788-5 
• Rudin, Walter (1991), Functional Analysis, McGraw-Hill Science, ISBN 0-07-054236-8 
• Willard, Stephen (February 2004), General Topology, Courier Dover Publications, ISBN 0-486-43479-6 
• Yosida, Kosaku (1980), Functional analysis (6th ed.), Springer, ISBN 978-3540586548 

See also

• Weak convergence (Hilbert space)
• Weak-star operator topology
• Topologies on the set of operators on a Hilbert space
• Vague topology