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 ${\displaystyle (x_{n})}$ in a Hilbert space H is said to converge weakly to a point x in H if

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

for all y in H. Here, ${\displaystyle \langle \cdot ,\cdot \rangle }$ is understood to be the inner product on the Hilbert space. The notation

${\displaystyle x_{n}\rightharpoonup x}$

is sometimes used to denote this kind of convergence.^[1]

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 ${\displaystyle x_{n}}$ 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 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 ${\displaystyle x_{n}}$ converges weakly to x, then

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

and this inequality is strict whenever the convergence is not strong. For example, infinite orthonormal sequences converge weakly to zero, as demonstrated below.

• If ${\displaystyle x_{n}\to x}$ weakly and ${\displaystyle \lVert x_{n}\rVert \to \lVert x\rVert }$, then ${\displaystyle x_{n}\to x}$ strongly:

${\displaystyle \langle x-x_{n},x-x_{n}\rangle =\langle x,x\rangle +\langle x_{n},x_{n}\rangle -\langle x_{n},x\rangle -\langle x,x_{n}\rangle \rightarrow 0.}$

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

Example

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

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 L^p 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, which is by Riemann–Lebesgue lemma, 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 x ∈ H, we have

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

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

${\displaystyle |\langle e_{n},x\rangle |^{2}\rightarrow 0}$ (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

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

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

In the case where ${\displaystyle B}$ is a Hilbert space, then, by the Riesz representation theorem, $${\displaystyle f(\cdot )=\langle \cdot ,y\rangle }$$ for some ${\displaystyle y}$ in ${\displaystyle B}$, 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

References

1. "redirect". dept.math.lsa.umich.edu. Retrieved 2024-09-17.