Weak convergence (Hilbert space)

(Redirected from Banach-Saks theorem)

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

Definition

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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.[1]

Properties

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  • If a sequence converges strongly (that is, if it converges in norm), 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 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   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   weakly and  , then   strongly:
 
  • If the Hilbert space is finite-dimensional, i.e. a Euclidean space, then weak and strong convergence are equivalent.

Example

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The first three 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, 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."

Weak convergence of orthonormal sequences

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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

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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

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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 an Lp space on   and  , then any such   has the form   for some  , where   is the measure on   and   are conjugate indices.

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.

See also

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References

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  1. ^ "redirect". dept.math.lsa.umich.edu. Retrieved 2024-09-17.