Applied linear operators and spectral methods/Weak convergence

Convergence of distributions
 Definition:

A sequence of distributions $$\{t_n\}$$ is said to converge to the distribution $$t$$ if their actions converge in $$\mathbb{R}$$, i.e.,

\left\langle t_n, \phi \right\rangle \rightarrow \left\langle t, \phi \right\rangle \qquad \forall~\phi \in D \quad \text{as}~b \rightarrow \infty $$ This is called  convergence in the sense of distributions or  weak convergence.

For example,

t_n := \left\langle \sin(nx), \phi \right\rangle = \int_{-\infty}^{\infty} \sin(nx)~\phi(x)~dx \rightarrow 0 \qquad \text{as} n \rightarrow \infty $$ Therefore, $$\{\sin(nx)\}$$ converges to $$\sin(0)$$ as $$n \rightarrow \infty$$, in the weak sense of distributions.

If $$\{t_n\} \rightarrow t$$ if follows that the derivatives $$\{t'_n\}$$ will converge to $$t'$$ since

\left\langle t'_n, \phi \right\rangle = -\left\langle t_n, \phi' \right\rangle \rightarrow -\left\langle t, \phi' \right\rangle = \left\langle t', \phi \right\rangle $$

For example, $$\{t_n\} = \left\{\cfrac{\cos(nx)}{n}\right\}$$ is both a sequence of functions and a sequence of distributions which, as $$n \rightarrow \infty$$, converge to 0 both as a function (i.e., pointwise or in $$L^2$$) or as a distribution.

Also, $$\{t'_n\} = \{-\sin(nx)\}$$ converges to the zero distribution even though its pointwise limit is not defined.