A sequence of distributions is said to converge to the distribution
if their actions converge in , i.e.,
This is called convergence in the sense of distributions or
weak convergence.
For example,
Therefore, converges to as , in
the weak sense of distributions.
If if follows that the derivatives will
converge to since
For example, is both a
sequence of functions and a sequence of distributions which, as
, converge to 0 both as a function (i.e., pointwise or
in ) or as a distribution.
Also, converges to the zero distribution even though
its pointwise limit is not defined.
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