I am working through Reed and Simon's book on functional analysis. This is one of their exercises:
Let $\mathscr{S}$ denote the space of Schwartz functions, $\mathscr{S}'$ the space of tempered distributions, and $\delta$ the delta function.
(a) Let $\varphi \in \mathscr{S}$ and let $\varphi_y$ be the function in $\mathscr{S}$ defined by $\varphi_y(x) = \varphi(x-y)$. Prove that the map $y \mapsto \varphi_y$ is a $C^\infty$ function from $\mathbb{R}^n$ to $\mathscr{S}(\mathbb{R}^n)$ with $D^\alpha(\varphi_y) = (-1)^\alpha(D^\alpha \varphi)_y.$ To say $y \mapsto \varphi_y$ has derivative $\frac{\partial \varphi_y}{\partial y_j}$ as a function with values in $\mathscr{S}$ means $$\lim_{y\rightarrow y_0} |y-y_0|^{-1} \Big[\varphi_y - \varphi_{y_0} - \sum_{j=0}^N \frac\partial{\partial y_j}(\varphi_y) \cdot (y-y_0)_j\Big] = 0$$ in the topology of $\mathscr{S}$.
(b) Let $T \in \mathscr{S}'$. Let $\varphi \in \mathscr{S}$. Define $T^\varphi$ to be the function, $T^\varphi(y) = T(\varphi_y)$. Prove that $T^\varphi \in C^\infty$.
(c) Let $\varphi_n \in \mathscr{S}$ with $\varphi_n \rightarrow \delta$ in the weak topology on $\mathscr{S}'$. Prove that $T^{\varphi_n} \rightarrow T$ for all $T \in \mathscr{S}'$ in the weak topology on $\mathscr{S}'$.
(d) Prove that $\mathscr{S}$ is dense in $\mathscr{S}'$.
I could not find a similar problem so I am having some trouble getting started. I know that $\mathscr{S}$ is generated by the seminorms $$\|\varphi\|_{\alpha, \beta} = \sup_{x \in \mathbb{R}^n} |x^\alpha D^\beta\varphi(x)|$$ but not much beyond that. I would like to use this exercise as a learning experience for tempered distributions. Any help on how to approach these problems would be much appreciated.
As suggested in the comments, these are the definitions I am using:
- A map $f: \mathbb{R}^n \rightarrow \mathbb{C}$ is in $\mathscr{S}(\mathbb{R}^n)$ if $\|f\|_{\alpha, \beta} < \infty$ for all $\alpha$ and $\beta$ (which are $n$-tuples of nonnegative integers).
- A tempered distribution $T \in \mathscr{S}'$ is a continuous linear functional on $\mathscr{S}$. Thus $T$ is continuous if and only if there exists a $i_1, \ldots, i_n$ and constants $c_1, \ldots, c_n$ such that $$\forall f \in \mathscr{S}: \quad |T(f)| \leq \sum_{j=1}^{n} c_{j} \| f \|_{i_{j}}.$$