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| {{about|the weak topology on a normed vector space|the weak topology induced by a family of maps|initial topology|the weak topology generated by a cover of a space|coherent topology}}
| | Greetings! I am Myrtle Shroyer. For a whilst she's been in South Dakota. I am a meter reader but I plan on altering it. To gather cash is one of the issues I love most.<br><br>my homepage - [http://Yzu.ch/weightlossfoodprograms67803 diet meal delivery] |
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| In [[mathematics]], '''weak topology''' is an alternative term for [[initial topology]]. 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]].
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| 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 '''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.
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| ==The weak and strong topologies==
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| Let ''K'' 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 ''K'' will be either the field of [[complex numbers]] or the field of [[real number]]s with the familiar topologies. Let ''X'' be a [[topological vector space]] over ''K''. Namely, ''X'' is a ''K'' [[vector space]] equipped with a [[topological space | topology]] so that vector addition and [[scalar multiplication]] are continuous.
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| We may define a possibly different topology on ''X'' using the continuous (or ''topological'') [[dual space]] ''X<sup>*</sup>''. The topological dual space consists of all [[linear operator|linear functions]] from ''X'' into the base field ''K'' which are [[continuous function (topology)|continuous]] with respect to the given topology. The '''weak topology''' on ''X'' is the [[initial topology]]
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| with respect to ''X<sup>*</sup>''. In other words, it is the [[comparison of topologies | coarsest]] topology (the topology with the fewest open sets) such that each element of ''X<sup>*</sup>'' is a [[continuous function]]. In order to distinguish the weak topology from the original topology on ''X'', the original topology is often called the '''strong topology'''.
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| A [[subbase]] for the weak topology is the collection of sets of the form φ<sup>-1</sup>(''U'') where φ ∈ ''X''<sup>*</sup> 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 φ<sup>-1</sup>(''U'').
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| More generally, if ''F'' is a subset of the [[algebraic dual space]], then the [[initial topology]] of ''X'' with respect to ''F'', denoted by σ(''X'',''F''), is the '''weak topology with respect to ''F'' '''. If one takes ''F'' to be the whole continuous dual space of ''X'', then the weak topology with respect to ''F'' coincides with the weak topology defined above.
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| If the field ''K'' has an [[absolute value]] <math> |\cdot| </math>, then the weak topology σ(''X'',''F'') is induced by the family of
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| [[seminorm]]s,
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| :<math>\|x\|_f \overset{\text{def}}{=} |f(x)|</math>
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| for all ''f''∈''F'' and ''x''∈''X''. In particular, weak topologies are [[locally convex space|locally convex]].
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| From this point of view, 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'' {{harv|Rudin|1991|loc=Theorem 3.10}}.
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| ===Weak convergence===
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| {{selfref|For further information, see: [[Weak convergence (Hilbert space)]]}}
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| The weak topology is characterized by the following condition: a [[net (mathematics)|net]] (''x''<sub>''λ''</sub>) in ''X'' converges in the weak topology to the element ''x'' of ''X'' if and only if φ(''x''<sub>''λ''</sub>) converges to φ(''x'') in '''R''' or '''C''' for all φ in ''X*'' .
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| In particular, if ''x''<sub>n</sub> is a [[sequence (mathematics)|sequence]] in ''X'', then ''x''<sub>n</sub> '''converges weakly to''' ''x'' if
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| :<math>\phi(x_n) \to \phi(x)</math>
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| as ''n'' → ∞ for all φ ∈ ''X''<sup>*</sup>. In this case, it is customary to write
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| :<math>x_n \overset{\mathrm{w}}{\longrightarrow} x</math>
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| or, sometimes,
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| :<math>x_n \rightharpoonup x.</math>
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| ===Other properties===
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| 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]].
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| If ''X'' is a normed space, then the dual space ''X*'' is itself a normed vector space by using the norm ǁφǁ = sup<sub>ǁxǁ≤1</sub>|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.
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| ==The weak-* topology==
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| <!-- weak* convergence in normed linear space links to this heading -->
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| A space ''X'' can be embedded into the [[double dual]] ''X**'' by
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| :<math>x \mapsto T_x</math>
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| where
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| :<math>T_x(\phi) = \phi(x).\ </math>
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| Thus ''T'' : ''X'' → ''X**'' 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 ''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<sub>x</sub>'', defined by ''T<sub>x</sub>(φ) = φ(x)'' from ''X*'' to the base field '''R''' or '''C''' remain continuous.
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| === Weak-* convergence ===
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| A [[net (mathematics)|net]] ''φ<sub>λ</sub>'' in ''X*'' is convergent to ''φ'' in the weak-* topology if it converges pointwise:
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| :<math>\phi_{\lambda} (x) \rightarrow \phi (x)</math>
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| for all ''x'' in ''X''. In particular, a [[sequence (mathematics)|sequence]] of φ<sub>n</sub> ∈ ''X''<sup>*</sup> converges to φ provided that
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| :<math>\phi_n(x)\to\phi(x)</math>
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| for all ''x'' in ''X''. In this case, one writes
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| :<math>\phi_n \overset{w^*}{\rightarrow} \phi</math>
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| as ''n'' → ∞.
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| 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.
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| === Other properties ===
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| 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 closed unit ball in ''X*'' is weak*-[[compact space|compact]] (more generally, the [[polar set|polar]] in ''X*'' 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 space|reflexive]].
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| 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 ''X''*, the topological dual space ''X'' of continuous ''F''-valued linear functionals on ''X'', all norm-closed balls are compact in the weak-* topology.
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| If a normed space ''X'' is separable, then the weak-* topology is metrizable on the norm-bounded subsets of ''X*''. It is not metrizable on all of ''X''* unless the normed space ''X'' is countable-dimensional. If ''X'' is a Banach space, the weak-* topology is not metrizable on all of ''X''* unless ''X'' is finite-dimensional.<ref>Proposition 2.6.12, p. 226 in {{citation
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| | last = Megginson | first = Robert E.
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| | title = An introduction to Banach space theory
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| | series = Graduate Texts in Mathematics
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| | volume = 183
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| | publisher = Springer-Verlag
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| | location = New York
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| | year = 1998
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| | pages = xx+596
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| | isbn = 0-387-98431-3
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| }}.</ref>
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| ==Examples==
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| ===Hilbert spaces===
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| Consider, for example, the difference between strong and weak convergence of functions in the [[Hilbert space]] [[Lp space|''L''<sup>2</sup>('''R'''<sup>n</sup>)]]. Strong convergence of a sequence ψ<sub>k</sup>∈''L''<sup>2</sup>('''R'''<sup>n</sup>) to an element ψ means that
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| :<math>\int_{\mathbf{R}^n} |\psi_k-\psi |^2\,{\rm d}\mu\, \to 0\,</math>
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| as ''k''→∞. Here the notion of convergence corresponds to the norm on ''L''<sup>2</sup>.
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| In contrast weak convergence only demands that
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| :<math>\int_{\mathbf{R}^n} \bar{\psi}_k f\,\mathrm d\mu \to \int_{\mathbf{R}^n} \bar{\psi}f\, \mathrm d\mu</math>
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| for all functions ''f''∈''L''<sup>2</sup> (or, more typically, all ''f'' in a [[dense subset]] of ''L''<sup>2</sup> such as a space of [[test function]]s, if the sequence {''ψ''<sub>''k''</sub>} is bounded). For given test functions, the relevant notion of convergence only corresponds to the topology used in '''C'''.
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| For example, in the Hilbert space ''L''<sup>2</sup>(0,π), the sequence of functions
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| :<math>\psi_k(x) = \sqrt{2/\pi}\sin(k x)</math> | |
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| form an [[orthonormal basis]]. In particular, the (strong) limit of ψ<sub>k</sub> as ''k''→∞ does not exist. On the other hand, by the [[Riemann–Lebesgue lemma]], the weak limit exists and is zero.
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| ===Distributions===
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| 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 '''R'''<sup>n</sup>). 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 ''L''<sup>2</sup>. Thus one is led to consider the idea of a [[rigged Hilbert space]].
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| ==Operator topologies==
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| If ''X'' and ''Y'' are topological vector spaces, the space ''L''(''X'',''Y'') of [[continuous linear operator]]s ''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 {{harv|Yosida|1980|loc=IV.7 Topologies of linear maps}}. There are, in general, a vast array of possible [[operator topology|operator topologies]] on ''L''(''X'',''Y''), whose naming is not entirely intuitive.
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| 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'':
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| :<math>f\mapsto \|f(x)\|_Y.</math>
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| More generally, if a family of seminorms ''Q'' defines the topology on ''Y'', then the seminorms ''p''<sub>q,x</sub> on ''L''(''X'',''Y'') defining the strong topology are given by
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| :<math>p_{q,x} : f \mapsto q(f(x)),</math>
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| indexed by ''q''∈''Q'' and ''x''∈''X''.
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| In particular, see the [[weak operator topology]] and [[weak* operator topology]].
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| == See also ==
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| *[[Eberlein compactum]], a compact set in the weak topology
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| *[[Weak convergence (Hilbert space)]]
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| *[[Weak-star operator topology]]
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| *[[Convergence_of_measures#Weak_convergence_of_measures|Weak convergence of measures]]
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| *[[Topologies on the set of operators on a Hilbert space]]
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| *[[Vague topology]]
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| == Notes ==
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| {{reflist}}
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| ==References==
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| * {{citation|last=Conway|first=John B.|title=A Course in Functional Analysis|edition=2nd|publisher=Springer-Verlag|year=1994|isbn=0-387-97245-5}}
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| * {{citation |last=Pedersen |first=Gert |title=Analysis Now |year=1989 |publisher=Springer |isbn=0-387-96788-5}}
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| * {{Cite isbn|9780070542365}} <!-- Rudin, Walter's Functional Analysis -->
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| * {{Cite isbn|9780486434797}} <!-- Willard, Stephen's General Topology (2004) -->
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| * {{citation|last=Yosida|first=Kosaku|title=Functional analysis|publisher=Springer|isbn=978-3-540-58654-8|year=1980|edition=6th}}
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| {{Functional Analysis}}
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| [[Category:Topology of function spaces]]
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| [[Category:Topology]]
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| [[Category:General topology]]
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