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.

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 x_n 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.
\Vert x\Vert \le \liminf_{n\to\infty} \Vert x_n \Vert,
and this inequality is strict whenever the convergence is not strong. For example, infinite orthonormal sequences converge weakly to zero, as demonstrated below.
  • If x_n converges weakly to x and we have the additional assumption that \lVert x_n \rVert \to \lVert x \rVert, then  x_n converges to x strongly:
\langle x - x_n, x - x_n \rangle = \langle x, x \rangle + \langle x_n, x_n \rangle - \langle x_n, x \rangle - \langle x, x_n \rangle \rightarrow 0.
  • 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 f_n(x) = \sin(n x) on [0, 2 \pi]. As n \rightarrow \infty f_n converges weakly to f =0.

The Hilbert space L^2[0, 2\pi] is the space of the square-integrable functions on the interval [0, 2\pi] equipped with the inner product defined by

\langle f,g \rangle = \int_0^{2\pi} f(x)\cdot g(x)\,dx,

(see Lp space). The sequence of functions f_1, f_2, \ldots defined by

f_n(x) = \sin(n x)

converges weakly to the zero function in L^2[0, 2\pi], as the integral

\int_0^{2\pi} \sin(n x)\cdot g(x)\,dx.

tends to zero for any square-integrable function g on [0, 2\pi] when n goes to infinity, i.e.

\langle f_n,g \rangle \to \langle 0,g \rangle = 0.

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[edit]

Consider a sequence e_n which was constructed to be orthonormal, that is,

\langle e_n, e_m \rangle = \delta_{mn}

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 xH, we have

 \sum_n | \langle e_n, x \rangle |^2 \leq \| x \|^2 (Bessel's inequality)

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

 | \langle e_n, x \rangle |^2 \rightarrow 0

i.e.

 \langle e_n, x \rangle  \rightarrow 0 .

Banach–Saks theorem[edit]

The Banach–Saks theorem states that every bounded sequence x_n contains a subsequence x_{n_k} and a point x such that

\frac{1}{N}\sum_{k=1}^N x_{n_k}

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 (x_n) in a Banach space B is said to converge weakly to a point x in B if

f(x_n) \to f(x)

for any bounded linear functional f defined on B, that is, for any f in the dual space B'. If B is a Hilbert space, then, by the Riesz representation theorem, any such f has the form

f(\cdot)=\langle \cdot,y \rangle

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