Weak convergence (Hilbert space)

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In mathematics, weak convergence in a Hilbert space is convergence of a sequence of points in the weak topology.

Definition[edit]

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[edit]

  • 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.
  • The norm is (sequentially) weakly lower-semicontinuous: 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 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[edit]

The first 3 curves in the sequence fn=sin(nx)
The first 3 functions in the sequence on . As converges weakly to .

The Hilbert space is the space of the square-integrable functions on the interval 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 on when goes to infinity, 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."

Weak convergence of orthonormal sequences[edit]

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 xH, we have

(Bessel's inequality)

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

(since the series above converges, its corresponding sequence must go to zero)

i.e.

Banach–Saks theorem[edit]

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[edit]

The definition of weak convergence can be extended to Banach spaces. 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 defined on , that is, for any in the dual space If is a Hilbert space, then, by the Riesz representation theorem, any such has the form

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