Weak convergence (Hilbert space) explained

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\lim _{n\toinfty} \langle x _{n,y} \rangle = \langle x,y \rangle

for all y in H. Here,

\langle ⋅ , ⋅ \rangle is understood to be the inner product on the Hilbert space. The notationx _{n} \rightharpoonup x

is sometimes used to denote this kind of convergence.

Properties

• If a sequence converges strongly (that is, if it converges in norm), then it converges weakly as well.

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 \le \liminf _{n\toinfty} \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 → 0 .

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

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 ^{2 \pi} _{0} f(x) ⋅ g(x)dx, (see L^p space). The sequence of functions f _{1,} f _{2,} \ldots defined byf _{n(x)} = \sin(n x) converges weakly to the zero function in L ^{2[0,} 2 \pi] , as the integral\int ^{2 \pi} _{0} \sin(n x) ⋅ 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 is a Hilbert space basis. Therefore

| \langle e _{n,} x \rangle | ^{2} → 0 (since the series above converges, its corresponding sequence must go to zero)

i.e.

\langle e _{n,} x \rangle → 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(1)/(N) \sum ^{N} _{k =1} 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

(x _{n)} in a Banach space B is said to converge weakly to a point x in B if$$f(x\_n)\; \backslash to\; f(x)$$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< +inf ty , then any such f has the form$$f(x)\; =\; \backslash int\_\; x\backslash ,y\backslash ,d\backslash mu$$for some y\inL ^{q(\Omega)} , where \mu is the measure on \Omega and (1)/(+ p) (1)/(q) =1 are conjugate indices.

In the case where

B is a Hilbert space, then, by the Riesz representation theorem,$$f(\backslash cdot)\; =\; \backslash langle\; \backslash cdot,y\; \backslash rangle$$for some y in B , so one obtains the Hilbert space definition of weak convergence.

See also

• • Operator topologies – topologies on the set of operators on a Hilbert space