Tempered distributions at non-coinciding points and density of Schwartz functions

Question

In the previous question, I find that situation is much less favorable than expected…. So I add more details to focus on the specific case I have in mind.

Let us consider the Schwartz space $\mathcal{S}(\mathbb{R}^{mN})$ and denotes its elements as $f(x_1, \dotsc, x_N)$ with $x_1, \dotsc, x_N \in \mathbb{R}^m$. In the original paper Osterwalder and Schrader - Axioms for Euclidean Green's functions on Schwinger functions, the following subspace is introduced: \begin{equation} ^{0}\mathcal{S}(\mathbb{R}^{mN}):= \{ f \in \mathcal{S}(\mathbb{R}^{mN}) \mid f \text{ and all its derivatives vanish at $(x_1, \dotsc, x_N)$ whenever } x_i=x_j \text{ for some } 1 \leq i < j \leq N \} \end{equation}

Let $ ^{0}\mathcal{S}(\mathbb{R}^{mN})'$ denote the space of continuous linear functionals on $ ^{0}\mathcal{S}(\mathbb{R}^{mN})$.

Obviously, each $f \in \mathcal{S}(\mathbb{R}^{mN})$ defines an element $T_f \in {^{0}\mathcal{S}}(\mathbb{R}^{mN})'$ by the formula $T_f (g) := \int_{\mathbb{R}^{mN}} fg$ for $g \in {^{0}\mathcal{S}}(\mathbb{R}^{mN})$.

Now, my question is as follows:

Is the above mapping $f \to T_f$ injective?

Is $\mathcal{S}(\mathbb{R}^{mN})$ sequentially dense in ${^{0}\mathcal{S}}(\mathbb{R}^{mN})'$ w.r.t the weak$^*$ topology in the sense that for any $T \in {^{0}\mathcal{S}}(\mathbb{R}^{mN})'$, we can find a sequence $\{ f_l \} \subset \mathcal{S}(\mathbb{R}^{mN})$ with $T_{f_l}(g) \to T(g)$ as $l \to \infty$ for each $g \in {^{0}\mathcal{S}}(\mathbb{R}^{mN})$?

I hope that this question is more solid than the previous one.

Add) After all, I am looking for a space of "ordinary functions" injectively and densely embedded in ${^{0}\mathcal{S}}(\mathbb{R}^{mN})'$. If Schwartz space doesn't work, I hope to know if there is any other candidate.

Answer 1

$\newcommand\R{\Bbb R}\newcommand\S{\mathcal S}$Yes, the linear mapping $f\mapsto T_f$ is injective.

Indeed, suppose that $T_f=0$ for some $f\in\S(\R^{mN})$.

Consider the open set $$X:=\{x=(x_1,\dots,x_N)\in\Bbb R^{mN}\colon x_i\ne x_j \text{ if }i\ne j\}.$$

Take any $x\in X$. Then there is a sequence $(g_n)$ in $^0\S(\R^{mN})$ such that eventually (that is, for each large enough $n$) we have $g_n\ge0$, $g_n=0$ outside the open ball $B_x(1/n)$ of radius $1/n$ centered at $x$, and $\int g_n=1$. Then $0=T_f(g_n)=\int fg_n\to f(x)$.

So, $f=0$ on $X$. Therefore and because $f$ is continuous and because $X$ is dense in $\R^{mN}$, we have $f=0$ on $\R^{mN}$. So, the mapping $f\mapsto T_f$ is injective.