Convergent sequence in the topology of $\mathscr{S_n}$ but not in that of $\mathcal{D}\mathbb{(R^n)}$

Question

We know the topology of the space $\mathscr{S_n}$ of rapidly decreasing functions is strictly weaker than the topology of the test function space $\mathcal{D}\mathbb{(R^n)}$. So I wish to construct a specific sequence ${ϕ_m}$ in $\mathcal{D}\mathbb{(R^n)}$ such that $ϕ_m → 0$ in the topology of $\mathscr{S_n}$ but not in the topology of $\mathcal{D}\mathbb{(R^n)}$. Are there any suggestions?

Answer 1

To answer the question in the comments:

First see:

Doubt in understanding Space $D(\Omega)$

for what it means to define the topology of a vector space using a collection of seminorms.

Now let $\mathscr{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions of rapid decay on $\mathbb{R}^n$. The topology is defined by the countable family of seminorms $$ ||f||_{\alpha,k}=\sup_{x\in\mathbb{R}^n}\langle x\rangle^k|\partial^{\alpha}f(x)| $$ indexed by $k\in\mathbb{N}_0$ and $\alpha\in\mathbb{N}_0^n$.

Here $\partial^\alpha\phi_m(x)=\frac{1}{m^{|\alpha|+1}}\partial^\alpha \rho(x/m)$.

Suppose the support of $\rho$ is contained in $[-M,M]^n$, then

$$ ||\phi_m||_{\alpha,k}\le \frac{1}{m^{|\alpha|+1}}\times (1+nM^2)^{\frac{k}{2}} \times \sup_{x\in [-M,M]^n}|\partial^\alpha\rho(x)| $$ which goes to zero when $m\rightarrow \infty$. Since this true for all $k$ and $\alpha$, this means the sequence $\phi_m$ converges to zero in Schwartz space.