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

$\lim _{n\to \infty }\langle x_{n},y\rangle =\langle x,y\rangle$

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

$\Vert x\Vert \leq \liminf _{n\to \infty }\Vert x_{n}\Vert ,$

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 first three functions in the sequence $f_{n}(x)=\sin(nx)$ on $[0,2\pi ]$. As $n\rightarrow \infty$ $f_{n}$ 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

$\langle f,g\rangle =\int _{0}^{2\pi }f(x)\cdot g(x)\,dx,$

(see L^p space). The sequence of functions $f_{1},f_{2},\ldots$ defined by

$f_{n}(x)=\sin(nx)$

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.

$\langle f_{n},g\rangle \to \langle 0,g\rangle =0.$

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 $L_{\infty }$ 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,

$\langle e_{n},e_{m}\rangle =\delta _{mn}$

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

$\sum _{n}|\langle e_{n},x\rangle |^{2}\leq \|x\|^{2}$ (Bessel's inequality)

where equality holds when {e_n} 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 $x_{n_{k}}$ 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

See also: Weak topology and Weak topology (polar topology)

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 $$f(x)=\int _{\Omega }x\,y\,d\mu$$ for some $y\in \,L^{q}(\Omega )$, where $\mu$ is the measure on $\Omega$ 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, $$f(\cdot )=\langle \cdot ,y\rangle$$ for some $y$ in $B$, so one obtains the Hilbert space definition of weak convergence.

See also

• Dual topology
• Operator topologies – topologies on the set of operators on a Hilbert space