 2

The space of smooth functions $\mathcal{E}(X)=C^{\infty }(X)$ has the topology of uniform convergence of every derivative on every compact set $K⋐X$. Thus $f_j\to f$ exactly whenfor every $K$ and every multi-index $\alpha$. Its continuous dual ${\mathcal{E}}^{\prime }(X)$ is the compactly supported distribution space; convergence in the weak dual topology means pointwise convergence on every $f\in \mathcal{E}(X)$.