Weak topology

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.

One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology.

HistoryEdit

Starting in the early 1900s, David Hilbert and Marcel Riesz made extensive use of weak convergence. The early pioneers of functional analysis did not elevate norm convergence above weak convergence and oftentimes viewed weak convergence as preferable.[1] In 1929, Banach introduced weak convergence for normed spaces and also introduced the analogous weak-* convergence.[1] The weak topology is also called topologie faible and schwache Topologie.

The weak and strong topologiesEdit

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.

Weak topology with respect to a pairingEdit

Both the weak topology and the weak* topology are special cases of a more general construction for pairings, which we now describe. The benefit of this more general construction is that any definition or result proved for it applies to both the weak topology and the weak* topology, thereby making redundant the need for many definitions, theorem statements, and proofs. This is also the reason why the weak* topology is also frequently referred to as the "weak topology"; because it is just an instance of the weak topology in the setting of this more general construction.

Suppose (X, Y, b) is a pairing of vector spaces over a topological field 𝕂 (i.e. X and Y are vector spaces over 𝕂 and b : X × Y → 𝕂 is a bilinear map).

Notation. For all xX, let b(x, •) : Y → 𝕂 denote the linear functional on Y defined by yb(x, y). Similarly, for all yY, let b(•, y) : X → 𝕂 be defined by xb(x, y).
Definition. The weak topology on X induced by Y (and b) is the weakest topology on X, denoted by 𝜎(X, Y, b) or simply 𝜎(X, Y), making all maps b(•, y) : X → 𝕂 continuous, as y ranges over Y.[1]

The weak topology on Y is now automatically defined as described in the article Dual system. However, for clarity, we now repeat it.

Definition. The weak topology on Y induced by X (and b) is the weakest topology on Y, denoted by 𝜎(Y, X, b) or simply 𝜎(Y, X), making all maps b(x, •) : Y → 𝕂 continuous, as x ranges over X.[1]

If the field 𝕂 has an absolute value ||, then the weak topology 𝜎(X, Y, b) on X is induced by the family of seminorms, py : X → ℝ, defined by

py(x) := |b(x, y)|

for all yY and xX. This shows that weak topologies are locally convex.

Assumption. We will henceforth assume that 𝕂 is either the real numbers or the complex numbers .

Canonical dualityEdit

We now consider the special case where Y is a vector subspace of the algebraic dual space of X (i.e. a vector space of linear functionals on X).

There is a pairing, denoted by   or  , called the canonical pairing whose bilinear map   is the canonical evaluation map, defined by   for all   and  . Note in particular that   is just another way of denoting   i.e.  .

Assumption. If Y is a vector subspace of the algebraic dual space of X then we will assume that they are associated with the canonical pairing X, Y.

In this case, the weak topology on X (resp. the weak topology on Y), denoted by 𝜎(X,Y) (resp. by 𝜎(Y,Y)) is the weak topology on X (resp. on Y) with respect to the canonical pairing X, Y.

The topology σ(X,Y) is the initial topology of X with respect to Y.

If Y is a vector space of linear functionals on X, then the continuous dual of X with respect to the topology σ(X,Y) is precisely equal to Y.[1](Rudin 1991, Theorem 3.10)

The weak and weak* topologiesEdit

Let X be a topological vector space (TVS) over 𝕂, that is, X is a 𝕂 vector space equipped with a topology so that vector addition and scalar multiplication are continuous. We call the topology that X starts with the original, starting, or given topology (the reader is cautioned against using the terms "initial topology" and "strong topology" to refer to the original topology since these already have well-known meanings, so using them may cause confusion). We may define a possibly different topology on X using the topological or continuous dual space  , which consists of all linear functionals from X into the base field 𝕂 that are continuous with respect to the given topology.

Recall that   is the canonical evaluation map defined by   for all   and  , where in particular,  .

Definition. The weak topology on X is the weak topology on X with respect to the canonical pairing  . That is, it is the weakest topology on X making all maps   continuous, as   ranges over  .[1]
Definition: The weak topology on   is the weak topology on   with respect to the canonical pairing  . That is, it is the weakest topology on   making all maps   continuous, as x ranges over X.[1] This topology is also called the weak* topology.

We give alternative definitions below.

Weak topology induced by the continuous dual spaceEdit

Alternatively, the weak topology on a TVS X is the initial topology with respect to the family  . In other words, it is the coarsest topology on X such that each element of   remains a continuous function.

A subbase for the weak topology is the collection of sets of the form   where   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  .

From this point of view, the weak topology is the coarsest polar topology; see weak topology (polar topology) for details.

Weak convergenceEdit

The weak topology is characterized by the following condition: a net   in X converges in the weak topology to the element x of X if and only if   converges to   in or for all  .

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

 

as n → ∞ for all  . In this case, it is customary to write

 

or, sometimes,

 

Other propertiesEdit

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.

If X is a normed space, then the dual space   is itself a normed vector space by using the norm

 

This norm gives rise to a topology, called the strong topology, on  . This is the topology of uniform convergence. The uniform and strong topologies are generally different for other spaces of linear maps; see below.

Weak-* topologyEdit

The weak* topology is an important example of a polar topology.

A space X can be embedded into its double dual X** by

 

Thus   is an injective linear mapping, though not necessarily surjective (spaces for which this canonical embedding is surjective are called reflexive). The weak-* topology on   is the weak topology induced by the image of  . In other words, it is the coarsest topology such that the maps Tx, defined by   from   to the base field or remain continuous.

Weak-* convergence

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

 

for all  . In particular, a sequence of   converges to   provided that

 

for all xX. In this case, one writes

 

as n → ∞.

Weak-* convergence is sometimes called the simple convergence or the pointwise convergence. Indeed, it coincides with the pointwise convergence of linear functionals.

PropertiesEdit

If X is a separable (i.e. has a countable dense subset) locally convex space and H is a subset of its continuous dual space, then H endowed with the weak* (subspace) topology is a metrizable topological space.[1] If X is a separable metrizable locally convex space then the weak* topology on the continuous dual space of X is separable.[1]

Properties on normed spaces

By definition, the weak* topology is weaker than the weak topology on  . An important fact about the weak* topology is the Banach–Alaoglu theorem: if X is normed, then the closed unit ball in   is weak*-compact (more generally, the polar in   of a neighborhood of 0 in X is weak*-compact). Moreover, the closed unit ball in a normed space X is compact in the weak topology if and only if X is reflexive.

In more generality, let F be locally compact valued field (e.g., the reals, the complex numbers, or any of the p-adic number systems). Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in  , the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak-* topology.

If X is a normed space, then a subset of the continuous dual is weak* compact if and only if it is weak* closed and norm-bounded.[1] This implies, in particular, the when X is an infinite-dimensional normed space then the closed unit ball at the origin in the dual space of X does not contain any weak* neighborhood of 0.[1]

If X is a normed space, then X is separable if and only if the weak-* topology on the closed unit ball of   is metrizable,[1] in which case the weak* topology is metrizable on every the norm-bounded subsets of  . If a normed space X has a dual space that is separable (with respect to the dual-norm topology) then X is necessarily separable.[1] If X is a Banach space, the weak-* topology is not metrizable on all of   unless X is finite-dimensional.[2]

ExamplesEdit

Hilbert spacesEdit

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

 

as k → ∞. Here the notion of convergence corresponds to the norm on L2.

In contrast weak convergence only demands that

 

for all functions fL2 (or, more typically, all f in a dense subset of L2 such as a space of test functions, if the sequence {ψk} is bounded). For given test functions, the relevant notion of convergence only corresponds to the topology used in .

For example, in the Hilbert space L2(0,π), the sequence of functions

 

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

DistributionsEdit

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 n). 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 led to consider the idea of a rigged Hilbert space.

Weak topology induced by the algebraic dualEdit

Suppose that X is a vector space and X# is the algebraic dual space of X (i.e. the vector space of all linear functionals on X). If X is endowed with the weak topology induced by X# then the continuous dual space of X is X#, every bounded subset of X is contained in a finite-dimensional vector subspace of X, every vector subspace of X is closed and has a topological complement.[3]

Operator topologiesEdit

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 xX:

 

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 qQ and xX.

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

See alsoEdit

ReferencesEdit

  1. ^ a b c d e f g h i j k l m Narici & Beckenstein 2011, pp. 225-273.
  2. ^ Proposition 2.6.12, p. 226 in Megginson, Robert E. (1998), An introduction to Banach space theory, Graduate Texts in Mathematics, 183, New York: Springer-Verlag, pp. xx+596, ISBN 0-387-98431-3.
  3. ^ Trèves 2006, pp. 36, 201.

BibliographyEdit