I'm reading this article and, at the begining of it, the author says: "If $(f,g) := \int dx dy f(x) v(x,y) g(y)$ is a positive-definite bounded bilinear form on $L^{2}(\mathbb{R}^{3})$, there exist a unique gaussian measure $d\phi_{\nu}$ in $\mathcal{S}'(\mathbb{R}^{3})$ with covariance $v(x,y)$ and $$e^{-\frac{1}{2}(f,f)} = \int d\phi_{\nu}e^{i\phi(f)} \hspace{1cm} (1)$$ This follows from Minlos' [Theorem]. If $v$ is sufficiently regular, $\sup v(x,x)< \infty$ and continuity is necessary, then we can take $f(x) =\sqrt{\beta} \sum_{j=1}^{N}e_{j}\delta(x-x_{j})$ so that [...]".
My question is: Why is that well-justified? I agree with the afirmative concerning Minlos' Theorem but as I understand, (1) follows from the restriction of $(f,f)$ to $\mathcal{S}(\mathbb{R}^{3})$ (in order to use Minlos). Why can we take $f$ to be a Dirac delta if Dirac deltas are not in $\mathcal{S}(\mathbb{R}^{3})$?