24.15.173.94: implementation

2014-08-23T04:00:09Z

implementation

en>Xezbeth: Disambiguated: fluctuation → Statistical fluctuations

2013-08-16T12:52:09Z

Disambiguated: fluctuation → Statistical fluctuations

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			In [[mathematics]], '''weak convergence''' in a [[Hilbert space]] is [[Limit of a sequence\|convergence]] of a [[sequence]] of points in the [[weak topology]].

			==Properties==
			*If a sequence converges strongly, then it converges weakly as well.

			*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.

			*As a consequence of the [[Banach-Steinhaus Theorem\|principle of uniform boundedness]], every weakly convergent sequence is bounded.

			*The norm is (sequentially) weakly [[Lower semicontinuous\|lower-semicontinuous]]: if <math>x_n</math> converges weakly to ''x'', then

			::<math>\Vert x\Vert \le \liminf_{n\to\infty} \Vert x_n \Vert, </math>

			:and this inequality is strict whenever the convergence is not strong. For example, infinite orthonormal sequences converge weakly to zero, as demonstrated below.

			*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:

			::<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>

			*If the Hilbert space is finite-dimensional, i.e. a [[Euclidean space]], then the concepts of weak convergence and strong convergence are the same.

			=== Example ===
			[[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>.]]

			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
			:<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
			:<math>f_n(x) = \sin(n x)</math>
			converges weakly to the zero function in <math>L^2[0, 2\pi]</math>, as the integral
			:<math>\int_0^{2\pi} \sin(n x)\cdot g(x)\,dx.</math>
			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.
			:<math>\langle f_n,g \rangle \to \langle 0,g \rangle = 0.</math>
			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."

			===Weak convergence of orthonormal sequences===
			Consider a sequence <math>e_n</math> which was constructed to be orthonormal, that is,

			:<math>\langle e_n, e_m \rangle = \delta_{mn}</math>

			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

			:<math> \sum_n \| \langle e_n, x \rangle \|^2 \leq \\| x \\|^2</math> ([[Bessel's inequality]])

			where equality holds when {''e''<sub>''n''</sub>} is a Hilbert space basis. Therefore

			:<math> \| \langle e_n, x \rangle \|^2 \rightarrow 0</math>

			i.e.

			:<math> \langle e_n, x \rangle \rightarrow 0 .</math>

			==Banach–Saks theorem==
			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

			:<math>\frac{1}{N}\sum_{k=1}^N x_{n_k}</math>

			converges strongly to ''x'' as ''N'' goes to infinity.

			==Generalizations==
			{{See also\|Weak topology\|Weak topology (polar topology)}}

			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

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

			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

			:<math>f(\cdot)=\langle \cdot,y \rangle</math>

			for some <math>y</math> in <math>B</math>, so one obtains the Hilbert space definition of weak convergence.

			{{Functional Analysis}}

			{{DEFAULTSORT:Weak Convergence (Hilbert Space)}}
			[[Category:Hilbert space]]

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24.15.173.94: implementation

en>Xezbeth: Disambiguated: fluctuation → Statistical fluctuations

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