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.^[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 $x_{n}$ 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 3 curves in the sequence fn=sin(nx) — 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

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.

References

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

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

[1]

Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
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Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
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Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
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Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F

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