- Weak convergence (Hilbert space)
In
mathematics , weak convergence is a type ofconvergence of asequence of points in aHilbert space (and, more generally, in aBanach space ).Definition
A sequence of points x_n) 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 angle o langle x,y angle
for all "y" in "H". Here, langle cdot, cdot angle is understood to be the
scalar product on the Hilbert space. The notation:x_n ightharpoonup x
is sometimes used to denote this kind of convergence.
Weak topology
Weak convergence is in contrast to strong convergence or convergence in the norm, which is defined by
:Vert x_n -x Vert o 0
where Vert x Vert = sqrt {langle x,x angle} 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 fromSchwarz 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 angle o langle x, x angle and x_n | o |x|, then we have x_n - x | o 0 as n o 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 angle = langle x_n, T^* y angle ightarrow langle x, T^*y angle = langle Tx, y angle.
Properties
*Since every closed and bounded set is weakly relatively compact (under the weak topology, its closure 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.*As a consequence of the principle of uniform boundedness, every weakly convergent sequence is bounded.
*If x_n converges weakly to "x", then
:Vert xVert le liminf_{n oinfty} 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 lim ||"xn"|| = ||"x"||, then "xn" converges to "x" strongly:
:langle x - x_n, x - x_n angle = langle x, x angle + langle x_n, x_n angle - langle x_n, x angle - langle x, x_n angle ightarrow 0.
Weak convergence of orthonormal sequences
Consider a sequence e_n which was constructed to be orthonormal, that is,
:langle e_n, e_m angle = 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 "x" ∈ "H", we have
:sum_n | langle e_n, x angle |^2 leq | x |^2
where equality holds when {"e""n"} is a Hilbert space basis. Therefore
:langle e_n, x angle |^2 ightarrow 0
i.e.
:langle e_n, x angle ightarrow 0 .
Banach-Saks theorem
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
The definition of weak convergence can be extended to
Banach space s. 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) o f(x)
for any scalar-valued
bounded linear operator f defined on B, that is, for any f in thedual space B'. If B is a Hilbert space, then, by theRiesz representation theorem , any such f has the form:f(x)=langle x,y angle
for some y in B, so one obtains the Hilbert space definition of weak convergence.
Wikimedia Foundation. 2010.