# Weak convergence (Hilbert space)

{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In mathematics, weak convergence in a Hilbert space is convergence of a sequence of points in the weak topology.

## Properties

• If a sequence converges strongly, then it converges weakly as well.
$\Vert x\Vert \leq \liminf _{n\to \infty }\Vert x_{n}\Vert ,$ and this inequality is strict whenever the convergence is not strong. For example, infinite orthonormal sequences converge weakly to zero, as demonstrated below.
$\langle x-x_{n},x-x_{n}\rangle =\langle x,x\rangle +\langle x_{n},x_{n}\rangle -\langle x_{n},x\rangle -\langle x,x_{n}\rangle \rightarrow 0.$ • If the Hilbert space is finite-dimensional, i.e. a Euclidean space, then the concepts of weak convergence and strong convergence are the same.

### Example

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

$\langle f,g\rangle =\int _{0}^{2\pi }f(x)\cdot g(x)\,dx,$ $f_{n}(x)=\sin(nx)$ converges weakly to the zero function in $L^{2}[0,2\pi ]$ , as the integral

$\int _{0}^{2\pi }\sin(nx)\cdot g(x)\,dx.$ $\langle f_{n},g\rangle \to \langle 0,g\rangle =0.$ ### 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 xH, we have

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

where equality holds when {en} 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

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
$f(x_{n})\to f(x)$ $f(\cdot )=\langle \cdot ,y\rangle$ 