View source for Weak convergence (Hilbert space)

{{Short description|Type of convergence in Hilbert spaces}}
{{onesource|date=December 2009}}
In [[mathematics]], '''weak convergence''' in a [[Hilbert space]] is [[Limit of a sequence|the convergence]] of a [[sequence]] of points in the [[weak topology]].

==Definition==
A [[sequence (mathematics)|sequence]] of points <math>(x_n)</math> in a Hilbert space ''H'' is said to '''converge weakly''' to a point ''x'' in ''H'' if

:<math>\lim_{n\to\infty}\langle x_n,y \rangle = \langle x,y \rangle</math>

for all ''y'' in ''H''.  Here, <math>\langle \cdot, \cdot \rangle</math> is understood to be the [[inner product]] on the Hilbert space. The notation

:<math>x_n \rightharpoonup x</math>

is sometimes used to denote this kind of convergence.<ref>{{Cite web |title=redirect |url=https://dept.math.lsa.umich.edu/ |access-date=2024-09-17 |website=dept.math.lsa.umich.edu}}</ref>

==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 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 infinite-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 \to x</math> weakly and <math>\lVert x_n \rVert \to \lVert x \rVert</math>, then <math> x_n \to 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 weak and strong convergence are equivalent.

===Example===
[[Image:Sinfrequency.jpg|alt=The first 3 curves in the sequence fn=sin(nx)|thumb|350px|The first three 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, which is by [[Riemann–Lebesgue lemma]],  i.e.
:<math>\langle f_n,g \rangle \to \langle 0,g \rangle = 0.</math>
Although <math>f_n</math> has an increasing number of 0's in <math>[0,2 \pi]</math> as <math>n</math> goes to infinity, it is of course not equal to the zero function for any <math>n</math>. Note that <math>f_n</math> does not converge to 0 in the <math>L_\infty</math> or <math>L_2</math> 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 <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> (since the series above converges, its corresponding sequence must go to zero)

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 display=block>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 an [[Lp space]] on <math>\Omega</math> and <math>p<+\infty</math>, then any such <math>f</math> has the form
<math display=block>f(x) = \int_{\Omega} x\,y\,d\mu</math>
for some <math>y\in\,L^q(\Omega)</math>, where <math>\mu</math> is the [[Measure (mathematics)|measure]] on <math>\Omega</math> and <math>\frac{1}{p}+\frac{1}{q}=1</math> are [[conjugate indices]].

In the case where <math>B</math> is a Hilbert space, then, by the [[Riesz representation theorem]],
<math display=block>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.

==See also==

* {{annotated link|Dual topology}}
* [[Operator topologies]] – topologies on the set of operators on a Hilbert space

==References==

{{reflist}}
{{reflist|group=note}}

{{Banach spaces}}
{{Hilbert space}}
{{Functional analysis}}

{{DEFAULTSORT:Weak Convergence (Hilbert Space)}}

[[Category:Convergence (mathematics)]]
[[Category:Hilbert spaces]]