weak convergence (Hilbert space)

In mathematics, weak convergence in a Hilbert space is the convergence of a sequence of points in the weak topology.

Definition

A sequence of points $(x_n)$ in a Hilbert space ''H'' is said to converge weakly to a point ''x'' in ''H'' if

:$\lim_\langle x_n,y \rangle = \langle x,y \rangle$

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 (its closure in the weak topology is compact), every bounded sequence $x_n$ 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 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 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)

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 spaces. 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 $f$ 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 $\frac+\frac=1$ are conjugate indices.

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

*
* Operator topologies – topologies on the set of operators on a Hilbert space

References

{{DEFAULTSORT:Weak Convergence (Hilbert Space)

Convergence (mathematics)
Hilbert spaces