Weak Convergence (Hilbert Space) - Generalizations

Generalizations

See also: Weak topology and Weak topology (polar topology)

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 defined on, that is, for any in the dual space If is a Hilbert space, then, by the Riesz representation theorem, any such has the form

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