In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, weak convergence in a Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

 is convergence

Limit of a sequence

The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

 of a sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

 of points in the weak topology

Weak topology

In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...

.

Definition

A sequence of points in a Hilbert space H is said to converge weakly to a point x in H if


for all y in H. Here, is understood to be the inner product on the Hilbert space. The notation


is sometimes used to denote this kind of convergence.

Weak topology

Weak convergence is in contrast to strong convergence or convergence in the norm, which is defined by


where is the norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

 of x.

The notion of weak convergence defines a topology on H and this is called the weak topology

Weak topology

In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...

 on H. In other words, the weak topology is the topology generated by the bounded functionals on H. It follows from Schwarz inequality that the weak topology is weaker than the norm topology. Therefore convergence in norm implies weak convergence while the converse is not true in general. However, if for each y and , then we have as

On the level of operators, a bounded operator T is also continuous in the weak topology: If x_n → x weakly, then for all y

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  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.

• If converges weakly to x, then

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

• If converges weakly to x and we have the additional assumption that lim ||x_n|| = ||x||, then x_n converges to x strongly:

• If the Hilbert space is of finite dimensional, i.e. a Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
  , then the concepts of weak convergence and strong convergence are the same.

Example

If for some , then converge weakly to 0 in whenever . (In the illustration the are simply integers.) It is clear that, while the function will be equal to zero more frequently n goes to infinity, it is not very similar to the zero function at all. 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 which was constructed to be orthonormal, that is,


where 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
(Bessel's inequality

Bessel's inequality

In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence....

)

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


i.e.

Banach-Saks theorem

The Banach-Saks theorem states that every bounded sequence contains a subsequence and a point x such that


converges strongly to x as N goes to infinity.

Generalizations

The definition of weak convergence can be extended to Banach space

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

s. A sequence of points in a Banach space B is said to converge weakly to a point x in B if


for any bounded linear functional

Functional (mathematics)

In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...

  defined on , that is, for any in the dual space

Dual space

In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...

  If is a Hilbert space, then, by the Riesz representation theorem

Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem. They are named in honour of Frigyes Riesz.- The Hilbert space representation theorem :...

, any such has the form


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