In
mathematics, weak convergence in a
Hilbert space is
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
* "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that united the four Wei ...
of a
sequence of points in the
weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
.
Definition
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
*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
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sinc ...
(its closure in the weak topology is compact), every
bounded sequence
In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that
:, f(x), \le M
for all ''x'' in ''X''. A ...
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
weakly and
, then
strongly:
::
*If the Hilbert space is finite-dimensional, i.e. a
Euclidean space, then weak and strong convergence are equivalent.
Example
The Hilbert space