# The convergence of distribution

Let $\mathcal S'(\mathbb R^n)$ the space of all continuous linear functions from the Schwartz space $\mathcal S(\mathbb R^n)$ to $\mathbb C$, and $\mathcal D'(\mathbb R^n)$ the space of all continuous linear functions from the space $\mathcal D(\mathbb R^n)$ of the smooth functions with compact support $\mathcal D(\mathbb R^n)$ to $\mathcal C$.

If ${f_j},f \in {\mathcal S}'({\mathbb R}^n)$ and ${f_j} \to f$ in ${\mathcal D}'(\mathbb R^n)$, is it necessarily that ${f_j} \to f$ in ${\mathcal S}'(\mathbb R^n)$?

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What is $S'$ and $D'$ here? – Paul Nov 22 '11 at 10:32

The answer is no. Consider the sequence $f_k=e^{nk^3}\delta_{(k,k,\ldots,k)}$. Since a function in $\mathcal D(\mathbb R^n)$ has a compact support, the sequence $\{f_k\}$ converges to $0$ in $\mathcal D'(\mathbb R^n)$. But we can have the convergence in $\mathcal S'(\mathbb R^n)$, since if we take $\varphi(x)=\exp\left(-\sum_{j=1}^nx_j^2\right)\in\mathcal S(\mathbb R^n)$, we have $$f_k(\varphi)=e^{nk^3}\exp\left(-\sum_{j=1}^n k^2\right)=\exp(n(k^3-k^2)),$$ which of course doesn't converge to $0$.
In fact, we can show that the sequence $a_k\delta_k$ converges in $\mathcal S'(\mathbb R)$ if and only if the sequence $\{a_k\}$ slowly increasing, namely, we can find $C>0$ and $N\in\mathbb N$ such that $|a_k|\leq C(1+k)^N$.