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.

History

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.Template:Sfn In 1929, Banach introduced weak convergence for normed spaces and also introduced the analogous weak-* convergence.Template:Sfn The weak topology is called {{#invoke:Lang|lang}} in French and {{#invoke:Lang|lang}} in German.

The weak and strong topologies

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Let <math>\mathbb{K}</math> be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications <math>\mathbb{K}</math> will be either the field of complex numbers or the field of real numbers with the familiar topologies.

Weak topology with respect to a pairing

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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 Template:Math is a pairing of vector spaces over a topological field <math>\mathbb{K}</math> (i.e. Template:Mvar and Template:Mvar are vector spaces over <math>\mathbb{K}</math> and Template:Math is a bilinear map).

Notation. For all Template:Math, let Template:Math denote the linear functional on Template:Mvar defined by Template:Math. Similarly, for all Template:Math, let Template:Math be defined by Template:Math.

Definition. The weak topology on Template:Mvar induced by Template:Mvar (and Template:Mvar) is the weakest topology on Template:Mvar, denoted by Template:Math or simply Template:Math, making all maps Template:Math continuous, as Template:Mvar ranges over Template:Mvar.Template:Sfn

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

Definition. The weak topology on Template:Mvar induced by Template:Mvar (and Template:Mvar) is the weakest topology on Template:Mvar, denoted by Template:Math or simply Template:Math, making all maps Template:Math continuous, as Template:Mvar ranges over Template:Mvar.Template:Sfn

If the field <math>\mathbb{K}</math> has an absolute value Template:Math, then the weak topology Template:Math on Template:Mvar is induced by the family of seminorms, Template:Math, defined by

Template:Math

for all Template:Math and Template:Math. This shows that weak topologies are locally convex.

Assumption. We will henceforth assume that <math>\mathbb{K}</math> is either the real numbers <math>\mathbb{R}</math> or the complex numbers <math>\mathbb{C}</math>.

Canonical duality

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

There is a pairing, denoted by <math>(X,Y,\langle\cdot, \cdot\rangle)</math> or <math>(X,Y)</math>, called the canonical pairing whose bilinear map <math>\langle\cdot, \cdot\rangle</math> is the canonical evaluation map, defined by <math>\langle x,x'\rangle =x'(x)</math> for all <math>x\in X</math> and <math>x'\in Y</math>. Note in particular that <math>\langle \cdot,x'\rangle</math> is just another way of denoting <math>x'</math> i.e. <math>\langle \cdot,x'\rangle=x'(\cdot)</math>.

Assumption. If Template:Mvar is a vector subspace of the algebraic dual space of Template:Mvar then we will assume that they are associated with the canonical pairing Template:Math.

In this case, the weak topology on Template:Mvar (resp. the weak topology on Template:Var), denoted by Template:Math (resp. by Template:Math) is the weak topology on Template:Mvar (resp. on Template:Mvar) with respect to the canonical pairing Template:Math.

The topology Template:Math is the initial topology of Template:Mvar with respect to Template:Mvar.

If Template:Mvar is a vector space of linear functionals on Template:Mvar, then the continuous dual of Template:Mvar with respect to the topology Template:Math is precisely equal to Template:Mvar.Template:SfnTemplate:Harv

The weak and weak* topologies

Let Template:Mvar be a topological vector space (TVS) over <math>\mathbb{K}</math>, that is, Template:Mvar is a <math>\mathbb{K}</math> vector space equipped with a topology so that vector addition and scalar multiplication are continuous. We call the topology that Template:Mvar 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 Template:Mvar using the topological or continuous dual space <math>X^*</math>, which consists of all linear functionals from Template:Mvar into the base field <math>\mathbb{K}</math> that are continuous with respect to the given topology.

Recall that <math>\langle\cdot,\cdot\rangle</math> is the canonical evaluation map defined by <math>\langle x,x'\rangle =x'(x)</math> for all <math>x\in X</math> and <math>x'\in X^*</math>, where in particular, <math>\langle \cdot,x'\rangle=x'(\cdot)= x'</math>.

Definition. The weak topology on Template:Mvar is the weak topology on Template:Mvar with respect to the canonical pairing <math>\langle X,X^*\rangle</math>. That is, it is the weakest topology on Template:Mvar making all maps <math>x' =\langle\cdot,x'\rangle:X\to\mathbb{K}</math> continuous, as <math>x'</math> ranges over <math>X^*</math>.Template:Sfn

Definition: The weak topology on <math>X^*</math> is the weak topology on <math>X^*</math> with respect to the canonical pairing <math>\langle X,X^*\rangle</math>. That is, it is the weakest topology on <math>X^*</math> making all maps <math>\langle x,\cdot\rangle:X^*\to\mathbb{K}</math> continuous, as Template:Mvar ranges over Template:Mvar.Template:Sfn This topology is also called the weak* topology.

We give alternative definitions below.

Weak topology induced by the continuous dual space

Alternatively, the weak topology on a TVS Template:Mvar is the initial topology with respect to the family <math>X^*</math>. In other words, it is the coarsest topology on X such that each element of <math>X^*</math> remains a continuous function.

A subbase for the weak topology is the collection of sets of the form <math>\phi^{-1}(U)</math> where <math>\phi\in X^*</math> and Template:Mvar is an open subset of the base field <math>\mathbb{K}</math>. In other words, a subset of Template:Mvar 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 <math>\phi^{-1}(U)</math>.

From this point of view, the weak topology is the coarsest polar topology.

Weak convergence

The weak topology is characterized by the following condition: a net <math>(x_\lambda)</math> in Template:Mvar converges in the weak topology to the element Template:Mvar of Template:Mvar if and only if <math>\phi(x_\lambda)</math> converges to <math>\phi(x)</math> in <math>\mathbb{R}</math> or <math>\mathbb{C}</math> for all <math>\phi\in X^*</math>.

In particular, if <math>x_n</math> is a sequence in Template:Mvar, then <math>x_n</math> converges weakly to Template:Mvar if

<math>\varphi(x_n) \to \varphi(x)</math>

as Template:Math for all <math>\varphi \in X^*</math>. In this case, it is customary to write

<math>x_n \overset{\mathrm{w}}{\longrightarrow} x</math>

or, sometimes,

<math>x_n \rightharpoonup x.</math>

Other properties

If Template:Mvar is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and Template:Mvar is a locally convex topological vector space.

If Template:Mvar is a normed space, then the dual space <math>X^*</math> is itself a normed vector space by using the norm

<math>\|\phi\|=\sup_{\|x\|\le 1} |\phi(x)|.</math>

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

Weak-* topology

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

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

<math>x \mapsto \begin{cases} T_x: X^* \to \mathbb{K} \\ T_x(\phi) = \phi(x) \end{cases}</math>

Thus <math>T:X\to X^{**}</math> is an injective linear mapping, though not necessarily surjective (spaces for which this canonical embedding is surjective are called reflexive). The weak-* topology on <math>X^*</math> is the weak topology induced by the image of <math>T:T(X)\subset X^{**}</math>. In other words, it is the coarsest topology such that the maps T_x, defined by <math>T_x(\phi)=\phi(x)</math> from <math>X^*</math> to the base field <math>\mathbb{R}</math> or <math>\mathbb{C}</math> remain continuous.

Weak-* convergence

A net <math>\phi_{\lambda}</math> in <math>X^*</math> is convergent to <math>\phi</math> in the weak-* topology if it converges pointwise:

<math>\phi_{\lambda} (x) \to \phi (x)</math>

for all <math>x\in X</math>. In particular, a sequence of <math>\phi_n\in X^*</math> converges to <math>\phi</math> provided that

<math>\phi_n(x)\to\phi(x)</math>

for all Template:Math. In this case, one writes

<math>\phi_n \overset{w^*}{\to} \phi</math>

as Template:Math.

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

Properties

If Template:Mvar is a separable (i.e. has a countable dense subset) locally convex space and H is a norm-bounded subset of its continuous dual space, then H endowed with the weak* (subspace) topology is a metrizable topological space.Template:Sfn However, for infinite-dimensional spaces, the metric cannot be translation-invariant.Template:Sfn If Template:Mvar is a separable metrizable locally convex space then the weak* topology on the continuous dual space of Template:Mvar is separable.Template:Sfn

Properties on normed spaces

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

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

If Template:Mvar is a normed space, a version of the Heine-Borel theorem holds. In particular, a subset of the continuous dual is weak* compact if and only if it is weak* closed and norm-bounded.Template:Sfn This implies, in particular, that when Template:Mvar is an infinite-dimensional normed space then the closed unit ball at the origin in the dual space of Template:Mvar does not contain any weak* neighborhood of 0 (since any such neighborhood is norm-unbounded).Template:Sfn Thus, even though norm-closed balls are compact, X* is not weak* locally compact.

If Template:Mvar is a normed space, then Template:Mvar is separable if and only if the weak* topology on the closed unit ball of <math>X^*</math> is metrizable,Template:Sfn in which case the weak* topology is metrizable on norm-bounded subsets of <math>X^*</math>. If a normed space Template:Mvar has a dual space that is separable (with respect to the dual-norm topology) then Template:Mvar is necessarily separable.Template:Sfn If Template:Mvar is a Banach space, the weak* topology is not metrizable on all of <math>X^*</math> unless Template:Mvar is finite-dimensional.<ref>Proposition 2.6.12, p. 226 in Template:Citation.</ref>

Examples

Hilbert spaces

Consider, for example, the difference between strong and weak convergence of functions in the Hilbert space Template:Math. Strong convergence of a sequence <math>\psi_k\in L^2(\R^n)</math> to an element Template:Mvar means that

<math>\int_{\R^n} |\psi_k-\psi |^2\,{\rm d}\mu\, \to 0</math>

as Template:Math. Here the notion of convergence corresponds to the norm on Template:Math.

In contrast weak convergence only demands that

<math>\int_{\R^n} \bar{\psi}_k f\,\mathrm d\mu \to \int_{\R^n} \bar{\psi}f\, \mathrm d\mu</math>

for all functions Template:Math (or, more typically, all f in a dense subset of Template:Math such as a space of test functions, if the sequence {ψ_k} is bounded). For given test functions, the relevant notion of convergence only corresponds to the topology used in <math>\mathbb{C}</math>.

For example, in the Hilbert space Template:Math, the sequence of functions

<math>\psi_k(x) = \sqrt{2/\pi}\sin(k x)</math>

form an orthonormal basis. In particular, the (strong) limit of <math>\psi_k</math> as Template:Math does not exist. On the other hand, by the Riemann–Lebesgue lemma, the weak limit exists and is zero.

Distributions

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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 <math>\mathbb{R}^n</math>). 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 Template:Math. Thus one is led to consider the idea of a rigged Hilbert space.

Weak topology induced by the algebraic dual

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

Operator topologies

If Template:Mvar and Template:Mvar are topological vector spaces, the space Template:Math of continuous linear operators Template:Math 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 Template:Mvar to define operator convergence Template:Harv. There are, in general, a vast array of possible operator topologies on Template:Math, whose naming is not entirely intuitive.

For example, the strong operator topology on Template:Math is the topology of pointwise convergence. For instance, if Template:Mvar is a normed space, then this topology is defined by the seminorms indexed by Template:Math:

<math>f\mapsto \|f(x)\|_Y.</math>

More generally, if a family of seminorms Q defines the topology on Template:Mvar, then the seminorms Template:Math on Template:Math defining the strong topology are given by

<math>p_{q,x} : f \mapsto q(f(x)),</math>

indexed by Template:Math and Template:Math.

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

See also

• Eberlein compactum, a compact set in the weak topology
• Weak convergence (Hilbert space)
• Weak-star operator topology
• Weak convergence of measures
• Topologies on spaces of linear maps
• Topologies on the set of operators on a Hilbert space
• Vague topology

Bibliography

• Template:Citation
• Template:Cite book
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Citation
• Template:Rudin Walter Functional Analysis
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels
• Template:Cite book
• Template:Citation