# 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.
• Since every closed and bounded set is weakly relatively compact (its closure in the weak topology is compact), every bounded sequence ${\displaystyle x_{n}}$ 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 infinitely 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.
${\displaystyle \Vert x\Vert \leq \liminf _{n\to \infty }\Vert x_{n}\Vert ,}$
and this inequality is strict whenever the convergence is not strong. For example, infinite orthonormal sequences converge weakly to zero, as demonstrated below.
${\displaystyle \langle x-x_{n},x-x_{n}\rangle =\langle x,x\rangle +\langle x_{n},x_{n}\rangle -\langle x_{n},x\rangle -\langle x,x_{n}\rangle \rightarrow 0.}$
• 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 ${\displaystyle L^{2}[0,2\pi ]}$ is the space of the square-integrable functions on the interval ${\displaystyle [0,2\pi ]}$ equipped with the inner product defined by

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

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

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

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

${\displaystyle \int _{0}^{2\pi }\sin(nx)\cdot g(x)\,dx.}$

tends to zero for any square-integrable function ${\displaystyle g}$ on ${\displaystyle [0,2\pi ]}$ when ${\displaystyle n}$ goes to infinity, i.e.

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

Although ${\displaystyle f_{n}}$ has an increasing number of 0's in ${\displaystyle [0,2\pi ]}$ as ${\displaystyle n}$ goes to infinity, it is of course not equal to the zero function for any ${\displaystyle n}$. Note that ${\displaystyle f_{n}}$ does not converge to 0 in the ${\displaystyle L_{\infty }}$ or ${\displaystyle L_{2}}$ 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 ${\displaystyle e_{n}}$ which was constructed to be orthonormal, that is,

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

where ${\displaystyle \delta _{mn}}$ 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

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

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

${\displaystyle |\langle e_{n},x\rangle |^{2}\rightarrow 0}$ (since the series above converges, its corresponding sequence must go to zero)

i.e.

${\displaystyle \langle e_{n},x\rangle \rightarrow 0.}$

## Banach–Saks theorem

The Banach–Saks theorem states that every bounded sequence ${\displaystyle x_{n}}$ contains a subsequence ${\displaystyle x_{n_{k}}}$ and a point x such that

${\displaystyle {\frac {1}{N}}\sum _{k=1}^{N}x_{n_{k}}}$

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 ${\displaystyle (x_{n})}$ in a Banach space B is said to converge weakly to a point x in B if
${\displaystyle f(x_{n})\to f(x)}$
for any bounded linear functional ${\displaystyle f}$ defined on ${\displaystyle B}$, that is, for any ${\displaystyle f}$ in the dual space ${\displaystyle B'.}$ If ${\displaystyle B}$ is a Hilbert space, then, by the Riesz representation theorem, any such ${\displaystyle f}$ has the form
${\displaystyle f(\cdot )=\langle \cdot ,y\rangle }$
for some ${\displaystyle y}$ in ${\displaystyle B}$, so one obtains the Hilbert space definition of weak convergence.