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The <space of smooth functions> $\mathcal E(X)=C^\infty(X)$ has the topology of uniform convergence of every derivative on every <compact set> $K\Subset X$. Thus $f_j\to f$ exactly when
$$
\sup_{x\in K}|\partial^\alpha(f_j-f)(x)|\longrightarrow0
$$
for every $K$ and every <multi-index> $\alpha$. Its <continuous dual space>[continuous dual] $\mathcal E'(X)$ is the <compactly supported distribution space>; convergence in the weak dual topology means pointwise convergence on every $f\in\mathcal E(X)$.
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