Weak convergence (Hilbert space)

Template:Short description Template:Onesource 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

$\underset{n\to \mathrm{\infty }}{lim}⟨x_n,y⟩=⟨x,y⟩$

for all $y$ in $H$. Here, $⟨\cdot ,\cdot ⟩$ 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

$\Vert x\Vert \le \underset{n\to \mathrm{\infty }}{lim\; inf}\Vert x_n\Vert ,$

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 $\Vert x_n\Vert \to \Vert x\Vert$, then $x_n\to x$ strongly:

$⟨x-x_n,x-x_n⟩=⟨x,x⟩+⟨x_n,x_n⟩-⟨x_n,x⟩-⟨x,x_n⟩\to 0.$

• If the Hilbert space is finite-dimensional, i.e. a Euclidean space, then weak and strong convergence are equivalent.

Example

The first three functions in the sequence $f_n(x)=\sin (nx)$ on $[0,2\pi ]$. As $n\to \mathrm{\infty }$ $f_n$ converges weakly to $f=0$.

The Hilbert space $L^2[0,2\pi ]$ is the space of the square-integrable functions on the interval $[0,2\pi ]$ equipped with the inner product defined by

$⟨f,g⟩={\int }_0^{2\pi }f(x)\cdot g(x)dx,$

(see L^p space). The sequence of functions $f_1,f_2,\dots$ defined by

$f_n(x)=\sin (nx)$

converges weakly to the zero function in $L^2[0,2\pi ]$, as the integral

${\int }_0^{2\pi }\sin (nx)\cdot g(x)dx.$

tends to zero for any square-integrable function $g$ on $[0,2\pi ]$ when $n$ goes to infinity, which is by Riemann–Lebesgue lemma, i.e.

$⟨f_n,g⟩\to ⟨0,g⟩=0.$

Although $f_n$ has an increasing number of 0's in $[0,2\pi ]$ 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 $L_{\mathrm{\infty }}$ 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,

$⟨e_n,e_m⟩={\delta }_{mn}$

where ${\delta }_{mn}$ 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

$\sum _n|⟨e_n,x⟩{|}^2\le \Vert x{\Vert }^2$ (Bessel's inequality)

where equality holds when {e_n} is a Hilbert space basis. Therefore

$|⟨e_n,x⟩{|}^2\to 0$ (since the series above converges, its corresponding sequence must go to zero)

i.e.

$⟨e_n,x⟩\to 0.$

Banach–Saks theorem

The Banach–Saks theorem states that every bounded sequence $x_n$ contains a subsequence $x_{n_k}$ and a point x such that

$\frac{1}{N}\sum _{k=1}^N{x}_{n_k}$

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 $B^{\prime }$. If $B$ is an Lp space on $\mathrm{\Omega }$ and $p<+\mathrm{\infty }$, then any such $f$ has the form $$f(x)={\int }_{\mathrm{\Omega }}xyd\mu$$ for some $y\in L^q(\mathrm{\Omega })$, where $\mu$ is the measure on $\mathrm{\Omega }$ and $\frac{1}{p}+\frac{1}{q}=1$ are conjugate indices.

In the case where $B$ is a Hilbert space, then, by the Riesz representation theorem, $$f(\cdot )=⟨\cdot ,y⟩$$ for some $y$ in $B$, so one obtains the Hilbert space definition of weak convergence.

See also

• Template:Annotated link
• Operator topologies – topologies on the set of operators on a Hilbert space