Space of smooth functions ($\mathcal{E}(X)$)

For an open set $X\subseteq {\mathbb{R}}^n$, the space $\mathcal{E}(X)=C^{\infty }(X)$ carries the Fréchet topology in which $f_j\to f$ exactly when every derivative converges uniformly on every compact set contained in $X$.

The continuous dual ${\mathcal{E}}^{\prime }(X)$ of $\mathcal{E}(X)$ is the space of compactly supported distributions on $X$. Its weak topology is pointwise convergence on $\mathcal{E}(X)$.