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.
Properties
If a sequence converges strongly, then it converges weakly as well.
The norm is weakly lower-semicontinuous: if converges weakly to x, then
If converges weakly to and we have the additional assumption that, then converges to strongly:
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 Hilbert space is the space of the square-integrable functions on the interval equipped with the inner product defined by . The sequence of functions defined by converges weakly to the zero function in, as the integral tends to zero for any square-integrable function on when goes to infinity, which is by Riemann–Lebesgue lemma, i.e. Although has an increasing number of 0's in as goes to infinity, it is of course not equal to the zero function for any. Note that does not converge to 0 in the or norms. This dissimilarity is one of the reasons why this type of convergence is considered to be "weak."
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 where equality holds when 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 spaces. A sequence of points in a Banach spaceB is said to converge weakly to a point x in B if for any bounded linear functional defined on, that is, for any in the dual space. If is an Lp space on, and then, any such has the form For some where and is the measure on. In the case where is a Hilbert space, then, by the Riesz representation theorem, for some in, so one obtains the Hilbert space definition of weak convergence.
