Weak convergence (Hilbert space)

In mathematics, weak convergence in a Hilbert space is 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

\langle x_n,y \rangle \to \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, 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 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.
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 $x_{n}$ converges weakly to $x$ and we have the additional assumption that $\lVert x_{n}\rVert \to \lVert x\rVert$ , then $x_{n}$ converges 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 the concepts of weak convergence and strong convergence are the same.

Example

The first 3 curves in the sequence fn=sin(nx)

The first 3 functions in the sequence

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

[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, 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 a Hilbert space, then, by the Riesz representation theorem, any such $f$ has the form

f(\cdot )=\langle \cdot ,y\rangle

for some $y$ in $B$ , so one obtains the Hilbert space definition of weak convergence.

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

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