Weak convergence (Hilbert space)

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In mathematics, weak convergence is a type of convergence of a sequence of points in a Hilbert space (and, more generally, in a Banach space).

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[edit] Definition

A sequence of points (xn) in a Hilbert space H, with n an integer, is said to converge weakly to a point x in H if

\langle x_n,y \rangle \to \langle x,y \rangle

for all y in H. Here, \langle \cdot, \cdot \rangle is understood to be the scalar product on the Hilbert space. The notation

x_n \rightharpoonup x

is sometimes used to denote this kind of convergence.

[edit] Weak topology

Weak convergence is in contrast to strong convergence or convergence in the norm, which is defined by

\Vert x_n -x \Vert \to 0

where \Vert x \Vert = \sqrt {\langle x,x \rangle} is the norm of x.

The notion of weak convergence defines a topology on H and this is called the weak topology on H. In other words, the weak topology is the topology generated by the bounded functionals on H. It follows from Schwarz inequality that the weak topology is weaker than the norm topology. Therefore convergence in norm implies weak convergence while the converse is not true in general. However, if \langle x_n, x \rangle \to \langle x, x \rangle and \| x_n \| \to \|x\|, then we have \| x_n - x \| \to 0 as n \to \infty.

On the level of operators, a bounded operator T is also continuous in the weak topology: If xn → x weakly, then for all y

\langle Tx_n, y \rangle = \langle x_n, T^* y \rangle \rightarrow \langle x, T^*y \rangle = \langle Tx, y \rangle.

[edit] Properties

  • Since every closed and bounded set is weakly relatively compact (under the weak topology, its closure is compact), every bounded sequence xn 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.
  • If xn converges weakly to x, then
\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 xn converges weakly to x and we have the additional assumption that lim ||xn|| = ||x||, then xn 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.

[edit] Weak convergence of orthonormal sequences

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

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

where δmn 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

\sum_n | \langle e_n, x \rangle |^2 \leq \| x \|^2

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 .

[edit] Banach-Saks theorem

The Banach-Saks theorem states that every bounded sequence xn 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.

[edit] Generalizations

The definition of weak convergence can be extended to Banach spaces. A sequence of points (xn) 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 scalar-valued bounded linear operator 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(x)=\langle x,y\rangle

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