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 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.

Properties

Example

The Hilbert space is the space of the square-integrable functions on the interval [0, 2\pi] equipped with the inner product defined by (see Lp space). The sequence of functions defined by converges weakly to the zero function in, as the integral 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. 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, 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 where equality holds when {en} is a Hilbert space basis. Therefore i.e.

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 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 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 for some, where \mu is the measure on \Omega and are conjugate indices. In the case where B is a Hilbert space, then, by the Riesz representation theorem, for some y in B, so one obtains the Hilbert space definition of weak convergence.