Weak convergence of tempered distributions
= Weak convergence of tempered distributions
A sequence $u_j\in\mathcal S'$ converges weakly to $u$ if $\langle u_j,\varphi\rangle\to\langle u,\varphi\rangle$ for every <Schwartz function> $\varphi$. This is the weak-star topology of the continuous dual of the <Schwartz space>, and differs from choosing a strong dual topology defined by uniform convergence on bounded sets of test functions.