Weak convergence of a sequence 
to a vector 
in a normed space 
means that
 |
for every continuous linear functional 
on 
.
It is commonly denoted 
.
In a Hilbert space, this condition is equivalent
to 
for every vector 
. Every strongly convergent sequence is weakly convergent, but the converse does not
hold in general.
For finite Borel measures 
and 
on a metric space 
, weak convergence means 
for every real function 
that is bounded and continuous. For probability
measures this is also called convergence in distribution
(Billingsley 1999). On a separable metric
space, the Prokhorov metric metrizes this
convergence of probability
measures.
