Definition (Schwartz space)

We define the Schwartz space as the space of smooth functions whose derivatives of all orders are rapidly decreasing: $$\mathcal{S}:=\left\{ \phi \in C^{\infty}(\mathbb{R};\mathbb{R}):\forall k,l\in \mathbb{Z}_{+},\,\sup_{t\in \mathbb{R}}\left|t^{l}\phi^{(k)}(t)\right|<\infty \right\}$$where $\phi^{(k)}(t)=\frac{d^{k}}{dt^{k}}\phi(t)$.

Definition (Convergence in Schwartz space)

Let $$p_{\alpha,\beta}(\phi):=\sup_{t\in \mathbb{R}} \left| t^{\alpha}\frac{d^{\beta}}{dt^{b}}\phi(t) \right| ,\quad\phi \in \mathcal{S},\,\alpha,\beta \in \mathbb{N}$$be a countable number of semi-norms. We say a sequence of Schwartz functions $\{ \phi_{n} \}_{n\in \mathbb{N}}\subset \mathcal{S}$ converge to another one $\phi \in \mathcal{S}$ if $$\lim_{ n \to \infty } p_{\alpha,\beta}(\phi_{n}-\phi)=0,\quad\alpha,\beta \in \mathbb{Z}_{+}$$

Definition (Schwartz metric)

With the above semi-norms we can define a metric. Let $\phi,\psi \in \mathcal{S}$ the Schwartz metric is $$d(\phi,\psi)=\sum_{n} \frac{1}{2^{n}} \frac{p_{n}(\phi)}{1+p_{n}(\phi)}$$where $n$ is a countable enumeration of the pairs $(\alpha,\beta)\in \mathbb{Z}_{+}^{2}$. (This definition looks off since the second element in the metric is not on the RHS….).