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In mathematics, weak convergence in a Hilbert space is
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen * "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that united the four Wei ...
of a sequence of points in the
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 ...
.


Definition

A sequence of points (x_n) in a Hilbert space ''H'' is said to converge weakly to a point ''x'' in ''H'' if :\langle x_n,y \rangle \to \langle x,y \rangle for all ''y'' in ''H''. Here, \langle \cdot, \cdot \rangle is understood to be the inner product on the Hilbert space. The notation :x_n \rightharpoonup x is sometimes used to denote this kind of convergence.


Properties

*If a sequence converges strongly (that is, if it converges in norm), then it converges weakly as well. *Since every closed and bounded set is weakly
relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sinc ...
(its closure in the weak topology is compact), every
bounded sequence In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that :, f(x), \le M for all ''x'' in ''X''. A ...
x_n 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 principle of uniform boundedness, every weakly convergent sequence is bounded. *The norm is (sequentially) weakly lower-semicontinuous: if x_n converges weakly to ''x'', then ::\Vert x\Vert \le \liminf_ \Vert x_n \Vert, :and this inequality is strict whenever the convergence is not strong. For example, infinite orthonormal sequences converge weakly to zero, as demonstrated below. *If x_n \to x weakly and \lVert x_n \rVert \to \lVert x \rVert, then x_n \to x strongly: ::\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. *If the Hilbert space is finite-dimensional, i.e. a Euclidean space, then weak and strong convergence are equivalent.


Example

The Hilbert space L^2 , 2\pi/math> is the space of the square-integrable functions on the interval , 2\pi/math> equipped with the inner product defined by :\langle f,g \rangle = \int_0^ f(x)\cdot g(x)\,dx, (see L''p'' space). The sequence of functions f_1, f_2, \ldots defined by :f_n(x) = \sin(n x) converges weakly to the zero function in L^2 , 2\pi/math>, as the integral :\int_0^ \sin(n x)\cdot g(x)\,dx. tends to zero for any square-integrable function g on , 2\pi/math> when n goes to infinity, which is by Riemann–Lebesgue lemma, i.e. :\langle f_n,g \rangle \to \langle 0,g \rangle = 0. Although f_n has an increasing number of 0's in ,2 \pi/math> as n goes to infinity, it is of course not equal to the zero function for any n. Note that f_n does not converge to 0 in the L_\infty or L_2 norms. 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 e_n which was constructed to be orthonormal, that is, :\langle e_n, e_m \rangle = \delta_ where \delta_ 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 : \sum_n , \langle e_n, x \rangle , ^2 \leq \, x \, ^2 (
Bessel's inequality In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828. Let H be a Hi ...
) where equality holds when is a Hilbert space basis. Therefore : , \langle e_n, x \rangle , ^2 \rightarrow 0 (since the series above converges, its corresponding sequence must go to zero) i.e. : \langle e_n, x \rangle \rightarrow 0 .


Banach–Saks theorem

The Banach–Saks theorem states that every bounded sequence x_n contains a subsequence x_ and a point ''x'' such that :\frac\sum_^N x_ converges strongly to ''x'' as ''N'' goes to infinity.


Generalizations

The definition of weak convergence can be extended to
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
s. A sequence of points (x_n) in a Banach space ''B'' is said to converge weakly to a point ''x'' in ''B'' if f(x_n) \to f(x) for any bounded linear
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
f defined on B, that is, for any f in the dual space B'. If B is an Lp space on \Omega and p<+\infty, then any such f has the form f(x) = \int_ x\,y\,d\mu for some y\in\,L^q(\Omega), where \mu is the measure on \Omega and \frac+\frac=1. In the case where B is a Hilbert space, then, by the Riesz representation theorem, f(\cdot) = \langle \cdot,y \rangle for some y in B, so one obtains the Hilbert space definition of weak convergence.


See also

* *


References

{{DEFAULTSORT:Weak Convergence (Hilbert Space) Convergence (mathematics) Hilbert space