Weak convergence (Hilbert space)

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

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] 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

goes to infinity, which is by Riemann–Lebesgue lemma, i.e.

has an increasing number of 0's in

goes to infinity, it is of course not equal to the zero function for any

This dissimilarity is one of the reasons why this type of convergence is considered to be "weak."

Consider a sequence

which was constructed to be orthonormal, that is, where

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

The Banach–Saks theorem states that every bounded sequence

and a point x such that converges strongly to x as N goes to infinity.

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

in the dual space

is an Lp space on

are conjugate indices.

is a Hilbert space, then, by the Riesz representation theorem,

, so one obtains the Hilbert space definition of weak convergence.