View source for Weak topology

{{Short description|Mathematical term}}
{{about|the weak topology on a normed vector space|the weak topology induced by a general family of maps|initial topology|the weak topology generated by a cover of a space|coherent topology}}

In [[mathematics]], '''weak topology''' is an alternative term for certain [[initial topology|initial topologies]], often on [[topological vector space]]s or spaces of [[linear operator]]s, 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 space|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 set|closed]] (respectively, [[compact set|compact]], etc.) with respect to the weak topology. Likewise, functions are sometimes called '''[[Dual system#Weak continuity|weakly continuous]]''' (respectively, '''weakly differentiable''', '''weakly analytic''', etc.) if they are [[continuous function|continuous]] (respectively, [[derivative|differentiable]], [[analytic function|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.{{sfn | Narici | Beckenstein | 2011 | pp=225–273}} In 1929, [[Stefan Banach|Banach]] introduced weak convergence for normed spaces and also introduced the analogous [[weak-* topology|weak-* convergence]].{{sfn | Narici | Beckenstein | 2011 | pp=225–273}} The weak topology is called {{lang|fr|topologie faible}} in French and {{lang|de|schwache Topologie}} in German.

== The weak and strong topologies ==
{{Main|Topologies on spaces of linear maps}}

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

=== Weak topology with respect to a pairing ===
{{Main|Dual system#Weak topology}}

Both the weak topology and the weak* topology are special cases of a more general construction for [[Dual system|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 {{math|(''X'', ''Y'', ''b'')}} is a [[Dual system|pairing]] of vector spaces over a topological field <math>\mathbb{K}</math> (i.e. {{mvar|X}} and {{mvar|Y}} are vector spaces over <math>\mathbb{K}</math> and {{math|''b'' : ''X'' × ''Y'' → <math>\mathbb{K}</math>}} is a [[bilinear map]]).

:'''Notation.''' For all {{math|''x'' ∈ ''X''}}, let {{math|''b''(''x'', •) : ''Y'' → <math>\mathbb{K}</math>}} denote the linear functional on {{mvar|Y}} defined by {{math|''y'' {{mapsto}} ''b''(''x'', ''y'')}}. Similarly, for all {{math|''y'' ∈ ''Y''}}, let {{math|''b''(•, ''y'') : ''X'' → <math>\mathbb{K}</math>}} be defined by {{math|''x'' {{mapsto}} ''b''(''x'', ''y'')}}.

:'''Definition.''' The '''weak topology on {{mvar|X}}''' induced by {{mvar|Y}} (and {{mvar|b}}) is the weakest topology on {{mvar|X}}, denoted by {{math|𝜎(''X'', ''Y'', ''b'')}} or simply {{math|𝜎(''X'', ''Y'')}}, making all maps {{math|''b''(•, ''y'') : ''X'' → <math>\mathbb{K}</math>}} continuous, as {{mvar|y}} ranges over {{mvar|Y}}.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}}

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

:'''Definition.''' The '''weak topology on {{mvar|Y}}''' induced by {{mvar|X}} (and {{mvar|b}}) is the weakest topology on {{mvar|Y}}, denoted by {{math|𝜎(''Y'', ''X'', ''b'')}} or simply {{math|𝜎(''Y'', ''X'')}}, making all maps {{math|''b''(''x'', •) : ''Y'' → <math>\mathbb{K}</math>}} continuous, as {{mvar|x}} ranges over {{mvar|X}}.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}}

If the field <math>\mathbb{K}</math> has an [[absolute value]] {{math|{{mabs|⋅}}}}, then the weak topology {{math|𝜎(''X'', ''Y'', ''b'')}} on {{mvar|X}} is induced by the family of [[seminorm]]s, {{math|''p''<sub>''y''</sub> : ''X'' → <math>\mathbb{R}</math>}}, defined by

:{{math|''p''<sub>''y''</sub>(''x'') :{{=}} {{mabs|''b''(''x'', ''y'')}}}}

for all {{math|''y'' &isin; ''Y''}} and {{math|''x'' &isin; ''X''}}. This shows that weak topologies are [[locally convex space|locally convex]].

:'''Assumption.''' We will henceforth assume that <math>\mathbb{K}</math> is either the [[real number]]s <math>\mathbb{R}</math> or the [[complex number]]s <math>\mathbb{C}</math>.

==== Canonical duality ====

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

There is a pairing, denoted by <math>(X,Y,\langle\cdot, \cdot\rangle)</math> or <math>(X,Y)</math>, called the [[Dual system#Canonical duality on a vector space|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 {{mvar|Y}} is a vector subspace of the [[algebraic dual space]] of {{mvar|X}} then we will assume that they are associated with the canonical pairing {{math|{{angbr|''X'', ''Y''}}}}.

In this case, the '''weak topology on {{mvar|X}}''' (resp. the '''weak topology on {{var|Y}}'''), denoted by {{math|𝜎(''X'',''Y'')}} (resp. by {{math|𝜎(''Y'',''X'')}}) is the [[Dual system#Weak topology|weak topology]] on {{mvar|X}} (resp. on {{mvar|Y}}) with respect to the canonical pairing {{math|{{angbr|''X'', ''Y''}}}}.

The topology {{math|σ(''X'',''Y'')}} is the [[initial topology]] of {{mvar|X}} with respect to {{mvar|Y}}.

If {{mvar|Y}} is a vector space of linear functionals on {{mvar|X}}, then the continuous dual of {{mvar|X}} with respect to the topology {{math|σ(''X'',''Y'')}} is precisely equal to {{mvar|Y}}.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}}{{harv|Rudin|1991|loc=Theorem 3.10}}

==== The weak and weak* topologies ====

Let {{mvar|X}} be a [[topological vector space]] (TVS) over <math>\mathbb{K}</math>, that is, {{mvar|X}} is a <math>\mathbb{K}</math> [[vector space]] equipped with a [[topological space|topology]] so that vector addition and [[scalar multiplication]] are continuous. We call the topology that {{mvar|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 {{mvar|X}} using the topological or [[continuous dual space]] <math>X^*</math>, which consists of all [[linear functional]]s from {{mvar|X}} into the base field <math>\mathbb{K}</math> that are [[continuous function (topology)|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 {{mvar|X}}''' is the weak topology on {{mvar|X}} with respect to the [[Dual system#Canonical duality on a vector space|canonical pairing]] <math>\langle X,X^*\rangle</math>. That is, it is the weakest topology on {{mvar|X}} making all maps <math>x' =\langle\cdot,x'\rangle:X\to\mathbb{K}</math> continuous, as <math>x'</math> ranges over <math>X^*</math>.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}}

:'''Definition''': The '''weak topology on <math>X^*</math>''' is the weak topology on <math>X^*</math> with respect to the [[Dual system#Canonical duality on a vector space|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 {{mvar|x}} ranges over {{mvar|X}}.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}} 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 {{mvar|X}} is the [[initial topology]] with respect to the family <math>X^*</math>. In other words, it is the [[comparison of topologies|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 {{mvar|U}} is an open subset of the base field <math>\mathbb{K}</math>. In other words, a subset of {{mvar|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 <math>\phi^{-1}(U)</math>.

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

=== Weak convergence ===
{{further|Weak convergence (Hilbert space)}}

The weak topology is characterized by the following condition: a [[net (mathematics)|net]] <math>(x_\lambda)</math> in {{mvar|X}} converges in the weak topology to the element {{mvar|x}} of {{mvar|X}} 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 (mathematics)|sequence]] in {{mvar|X}}, then <math>x_n</math> '''converges weakly to''' {{mvar|x}} if

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

as {{math|''n'' → ∞}} 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 {{mvar|X}} is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and {{mvar|X}} is a [[locally convex topological vector space]].

If {{mvar|X}} 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 ==
<!-- weak* convergence in normed linear space links to this heading -->
{{See also|Polar topology}}

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

A space {{mvar|X}} 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 space|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<sub>x</sub>'', 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 (mathematics)|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 (mathematics)|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 {{math|''x'' &isin; ''X''}}. In this case, one writes

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

as {{math|''n'' → ∞}}.

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

=== Properties ===

If {{mvar|X}} is a [[Separable space|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.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}} However, for infinite-dimensional spaces, the metric cannot be translation-invariant.{{sfn | Folland | 1999 | pp=170}} If {{mvar|X}} is a separable [[Metrizable TVS|metrizable]] [[locally convex]] space then the weak* topology on the continuous dual space of {{mvar|X}} is separable.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}}

;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 {{mvar|X}} is normed, then the closed unit ball in <math>X^*</math> is weak*-[[compact space|compact]] (more generally, the [[polar set|polar]] in <math>X^*</math> of a neighborhood of 0 in {{mvar|X}} is weak*-compact). Moreover, the closed unit ball in a normed space {{mvar|X}} is compact in the weak topology if and only if {{mvar|X}} is [[reflexive space|reflexive]].

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

If {{mvar|X}} 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.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}} This implies, in particular, that when {{mvar|X}} is an infinite-dimensional normed space then the closed unit ball at the origin in the dual space of {{mvar|X}} does not contain any weak* neighborhood of 0 (since any such neighborhood is norm-unbounded).{{sfn | Narici | Beckenstein | 2011 | pp=225-273}} Thus, even though norm-closed balls are compact, X* is not weak* [[locally compact space|locally compact]].

If {{mvar|X}} is a normed space, then {{mvar|X}} is separable if and only if the weak* topology on the closed unit ball of <math>X^*</math> is metrizable,{{sfn | Narici | Beckenstein | 2011 | pp=225-273}} in which case the weak* topology is metrizable on norm-bounded subsets of <math>X^*</math>. If a normed space {{mvar|X}} has a dual space that is separable (with respect to the dual-norm topology) then {{mvar|X}} is necessarily separable.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}} If {{mvar|X}} is a [[Banach space]], the weak* topology is not metrizable on all of <math>X^*</math> unless {{mvar|X}} is finite-dimensional.<ref>Proposition 2.6.12, p.&nbsp;226 in {{citation | last = Megginson | first = Robert E. | author-link = Robert Megginson | title = An introduction to Banach space theory | series = Graduate Texts in Mathematics | volume = 183 | publisher = Springer-Verlag | location = New York | year = 1998 | pages = xx+596 | isbn = 0-387-98431-3}}.</ref>

== Examples ==

=== Hilbert spaces ===

Consider, for example, the difference between strong and weak convergence of functions in the [[Hilbert space]] {{math|[[Lp space|''L''<sup>2</sup>(<math>\mathbb{R}^n</math>)]]}}. Strong convergence of a sequence <math>\psi_k\in L^2(\R^n)</math> to an element {{mvar|&psi;}} means that

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

as {{math|''k'' → ∞}}. Here the notion of convergence corresponds to the norm on {{math|''L''<sup>2</sup>}}.

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 {{math|''f'' &isin; ''L''<sup>2</sup>}} (or, more typically, all ''f'' in a [[dense subset]] of {{math|''L''<sup>2</sup>}} such as a space of [[test function]]s, if the sequence {''&psi;''<sub>''k''</sub>} 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 {{math|''L''<sup>2</sup>(0,π)}}, 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 {{math|''k'' → ∞}} does not exist. On the other hand, by the [[Riemann–Lebesgue lemma]], the weak limit exists and is zero.

=== Distributions ===
{{Main|distribution (mathematics)}}

One normally obtains spaces of [[distribution (mathematics)|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 {{math|''L''<sup>2</sup>}}. Thus one is led to consider the idea of a [[rigged Hilbert space]].

=== Weak topology induced by the algebraic dual ===

Suppose that {{mvar|X}} is a vector space and ''X''<sup>#</sup> is the [[algebraic dual]] space of {{mvar|X}} (i.e. the vector space of all linear functionals on {{mvar|X}}). If {{mvar|X}} is endowed with the weak topology induced by ''X''<sup>#</sup> then the continuous dual space of {{mvar|X}} is {{math|''X''<sup>#</sup>}}, every bounded subset of {{mvar|X}} is contained in a finite-dimensional vector subspace of {{mvar|X}}, every vector subspace of {{mvar|X}} is closed and has a [[topological complement]].{{sfn | Trèves | 2006 | pp=36, 201}}

==Operator topologies==

If {{mvar|X}} and {{mvar|''Y''}} are topological vector spaces, the space {{math|''L''(''X'',''Y'')}} of [[continuous linear operator]]s {{math|''f'' : ''X''&nbsp;→&nbsp;''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 {{mvar|''Y''}} to define operator convergence {{harv|Yosida|1980|loc=IV.7 Topologies of linear maps}}. There are, in general, a vast array of possible [[operator topology|operator topologies]] on {{math|''L''(''X'',''Y'')}}, whose naming is not entirely intuitive.

For example, the '''[[strong operator topology]]''' on {{math|''L''(''X'',''Y'')}} is the topology of ''pointwise convergence''. For instance, if {{mvar|''Y''}} is a normed space, then this topology is defined by the seminorms indexed by {{math|''x'' &isin; ''X''}}:

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

More generally, if a family of seminorms ''Q'' defines the topology on {{mvar|''Y''}}, then the seminorms {{math|''p''<sub>''q'', ''x''</sub>}} on {{math|''L''(''X'',''Y'')}} defining the strong topology are given by

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

indexed by {{math|''q'' &isin; ''Q''}} and {{math|''x'' &isin; ''X''}}.

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]]
* [[Convergence of measures#Weak convergence of measures|Weak convergence of measures]]
* [[Topologies on spaces of linear maps]]
* [[Topologies on the set of operators on a Hilbert space]]
* [[Vague topology]]

== References ==
{{reflist}}

==Bibliography==
* {{citation|last=Conway|first=John B.|title=A Course in Functional Analysis|edition=2nd|publisher=Springer-Verlag|year=1994|isbn=0-387-97245-5}}
* {{cite book |last=Folland |first=G.B. |title=Real Analysis: Modern Techniques and Their Applications |publisher=John Wiley & Sons, Inc. |year=1999 |isbn=978-0-471-31716-6 |edition=Second}} <!-- {{sfn | Folland | 1999 | p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | Beckenstein | 2011 | p=}} -->
* {{citation |last=Pedersen |first=Gert |title=Analysis Now |year=1989 |publisher=Springer |isbn=0-387-96788-5}}
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} -->
* {{Cite book | isbn = 9780486434797 | title = General Topology | last1 = Willard | first1 = Stephen | date = February 2004 | publisher = Courier Dover Publications }} <!-- Willard, Stephen's General Topology (2004) -->
* {{citation|last=Yosida|first=Kosaku|title=Functional analysis|publisher=Springer|isbn=978-3-540-58654-8|year=1980|edition=6th}}

{{Functional analysis}}
{{Duality and spaces of linear maps}}

[[Category:General topology]]
[[Category:Topology]]
[[Category:Topology of function spaces]]