Answer by Iosif Pinelis for Tempered distributions at non-coinciding points and density of Schwartz functions

$\newcommand\R{\Bbb R}\newcommand\S{\mathcal S}$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.