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| | == from His Holiness to the Lord of the universe == |
| In [[mathematics]], '''weak convergence''' in a [[Hilbert space]] is [[Limit of a sequence|convergence]] of a [[sequence]] of points in the [[weak topology]].
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| ==Properties==
| | In East Timor shrine strong clouds can only be regarded as common,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_46.htm オークリー サングラス 登山], is the 'main quasi Universe,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_29.htm ゴルフ サングラス オークリー],' a sacred place ...... because of the limited universe withstand East Timor,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_25.htm オークリー レディース サングラス], in the case of the presence of so many of the Lord of the universe,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_52.htm オークリー サングラス ジョウボーン], the universe they are a group of quasi- Lord simply can not break through,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_56.htm オークリーサングラス画像], not the supply of holy origin of the universe to break.<br><br>from His Holiness to the Lord of the universe,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_56.htm オークリーサングラス画像], the divine power consumption is extremely alarming. No source supply is impossible.<br><br>His Holiness ...... Treasure of the Sierra<br><br>from such an embarrassing primary identity quasi universe, all of a sudden become the most dazzling East Timor holy status of high ...... even more than any master universe,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_73.htm オークリー サングラス 人気], the universe strongest! East Timor ancestor shrine even met him personally, and he even said: 'rhino Wong Anatomy Board also proved your potential ...... even if you never get this heritage, and will also get my holy best cultivation. You do not need to try to pressure ......! '<br><br>'If the performance is good enough for you! unlimited potential,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_52.htm オークリー サングラス ジョウボーン]!' |
| *If a sequence converges strongly, then it converges weakly as well.
| | 相关的主题文章： |
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| *Since every closed and bounded set is weakly [[Relatively compact subspace|relatively compact]] (its closure in the weak topology is compact), every [[bounded sequence]] <math>x_n</math> in a Hilbert space ''H'' contains a weakly convergent subsequence. Note that closed and bounded sets are not in general weakly compact in Hilbert spaces (consider the set consisting of an [[orthonormal basis]] in an infinitely dimensional Hilbert space which is closed and bounded but not weakly compact since it doesn't contain 0). However, bounded and weakly closed sets are weakly compact so as a consequence every convex bounded closed set is weakly compact.
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| | | <li>[http://www.sofro.com.cn/plus/feedback.php?aid=1191 http://www.sofro.com.cn/plus/feedback.php?aid=1191]</li> |
| *As a consequence of the [[Banach-Steinhaus Theorem|principle of uniform boundedness]], every weakly convergent sequence is bounded.
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| | | <li>[http://lightsaberfightclub.com/v/activity http://lightsaberfightclub.com/v/activity]</li> |
| *The norm is (sequentially) weakly [[Lower semicontinuous|lower-semicontinuous]]: if <math>x_n</math> converges weakly to ''x'', then
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| | | <li>[http://mermaids.tw/forum/showthread.php?p=2571102#post2571102 http://mermaids.tw/forum/showthread.php?p=2571102#post2571102]</li> |
| ::<math>\Vert x\Vert \le \liminf_{n\to\infty} \Vert x_n \Vert, </math>
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| | | </ul> |
| :and this inequality is strict whenever the convergence is not strong. For example, infinite orthonormal sequences converge weakly to zero, as demonstrated below.
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| *If <math>x_n</math> converges weakly to <math>x</math> and we have the additional assumption that <math>\lVert x_n \rVert \to \lVert x \rVert</math>, then <math> x_n</math> converges to <math>x</math> strongly:
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| ::<math>\langle x - x_n, x - x_n \rangle = \langle x, x \rangle + \langle x_n, x_n \rangle - \langle x_n, x \rangle - \langle x, x_n \rangle \rightarrow 0.</math>
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| *If the Hilbert space is finite-dimensional, i.e. a [[Euclidean space]], then the concepts of weak convergence and strong convergence are the same.
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| === Example ===
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| [[Image:Sinfrequency.jpg|alt=The first 3 curves in the sequence fn=sin(nx)|thumb|350px|The first 3 functions in the sequence <math>f_n(x) = \sin(n x)</math> on <math>[0, 2 \pi]</math>. As <math>n \rightarrow \infty</math> <math>f_n</math> converges weakly to <math>f =0</math>.]]
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| The Hilbert space <math>L^2[0, 2\pi]</math> is the space of the [[square-integrable function]]s on the interval <math>[0, 2\pi]</math> equipped with the inner product defined by
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| :<math>\langle f,g \rangle = \int_0^{2\pi} f(x)\cdot g(x)\,dx,</math> | |
| (see [[Lp space|L<sup>''p''</sup> space]]). The sequence of functions <math>f_1, f_2, \ldots</math> defined by
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| :<math>f_n(x) = \sin(n x)</math>
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| converges weakly to the zero function in <math>L^2[0, 2\pi]</math>, as the integral
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| :<math>\int_0^{2\pi} \sin(n x)\cdot g(x)\,dx.</math>
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| tends to zero for any square-integrable function <math>g</math> on <math>[0, 2\pi]</math> when <math>n</math> goes to infinity, i.e.
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| :<math>\langle f_n,g \rangle \to \langle 0,g \rangle = 0.</math>
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| While <math>f_n</math> will be equal to zero more frequently as <math>n</math> goes to infinity, it is not very similar to the zero function at all. This dissimilarity is one of the reasons why this type of convergence is considered to be "weak."
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| ===Weak convergence of orthonormal sequences===
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| Consider a sequence <math>e_n</math> which was constructed to be orthonormal, that is,
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| :<math>\langle e_n, e_m \rangle = \delta_{mn}</math> | |
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| where <math>\delta_{mn}</math> equals one if ''m'' = ''n'' and zero otherwise. We claim that if the sequence is infinite, then it converges weakly to zero. A simple proof is as follows. For ''x'' ∈ ''H'', we have
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| :<math> \sum_n | \langle e_n, x \rangle |^2 \leq \| x \|^2</math> ([[Bessel's inequality]])
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| where equality holds when {''e''<sub>''n''</sub>} is a Hilbert space basis. Therefore
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| :<math> | \langle e_n, x \rangle |^2 \rightarrow 0</math> | |
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| i.e.
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| :<math> \langle e_n, x \rangle \rightarrow 0 .</math>
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| ==Banach–Saks theorem==
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| The '''Banach–Saks theorem''' states that every bounded sequence <math>x_n</math> contains a subsequence <math>x_{n_k}</math> and a point ''x'' such that
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| :<math>\frac{1}{N}\sum_{k=1}^N x_{n_k}</math> | |
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| converges strongly to ''x'' as ''N'' goes to infinity.
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| ==Generalizations==
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| {{See also|Weak topology|Weak topology (polar topology)}}
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| The definition of weak convergence can be extended to [[Banach space]]s. A sequence of points <math>(x_n)</math> in a Banach space ''B'' is said to '''converge weakly''' to a point ''x'' in ''B'' if
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| :<math>f(x_n) \to f(x)</math>
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| for any bounded linear [[functional (mathematics)|functional]] <math>f</math> defined on <math>B</math>, that is, for any <math>f</math> in the [[dual space]] <math>B'.</math> If <math>B</math> is a Hilbert space, then, by the [[Riesz representation theorem]], any such <math>f</math> has the form
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| :<math>f(\cdot)=\langle \cdot,y \rangle</math>
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| for some <math>y</math> in <math>B</math>, so one obtains the Hilbert space definition of weak convergence.
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| {{Functional Analysis}}
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| {{DEFAULTSORT:Weak Convergence (Hilbert Space)}}
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| [[Category:Hilbert space]]
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from His Holiness to the Lord of the universe
In East Timor shrine strong clouds can only be regarded as common,オークリー サングラス 登山, is the 'main quasi Universe,ゴルフ サングラス オークリー,' a sacred place ...... because of the limited universe withstand East Timor,オークリー レディース サングラス, in the case of the presence of so many of the Lord of the universe,オークリー サングラス ジョウボーン, the universe they are a group of quasi- Lord simply can not break through,オークリーサングラス画像, not the supply of holy origin of the universe to break.
from His Holiness to the Lord of the universe,オークリーサングラス画像, the divine power consumption is extremely alarming. No source supply is impossible.
His Holiness ...... Treasure of the Sierra
from such an embarrassing primary identity quasi universe, all of a sudden become the most dazzling East Timor holy status of high ...... even more than any master universe,オークリー サングラス 人気, the universe strongest! East Timor ancestor shrine even met him personally, and he even said: 'rhino Wong Anatomy Board also proved your potential ...... even if you never get this heritage, and will also get my holy best cultivation. You do not need to try to pressure ......! '
'If the performance is good enough for you! unlimited potential,オークリー サングラス ジョウボーン!'
相关的主题文章：