I already posted a similar question on MO and looked into the references therein.

However, I cannot find a satisfactory answer for my question..So I ask here again in a more refined form.

Let $T \in \mathcal{S}'(\mathbb{R}^{2n})$ be a tempered distribution. For $x, y \in \mathbb{R}^n$, we may write as $T(x,y)$ and understand that \begin{equation} T(f) = \int_{\mathbb{R}^n \times \mathbb{R}^n } T(x,y)f(x,y)d^nxd^ny \end{equation} for $f \in \mathcal{S}(\mathbb{R}^{2n})$.

Now, in the context of defining the Wick products, I frequently run into limits of the form \begin{equation} \lim\limits_{x \to y} T(x,y) \end{equation} I do not clearly see how to make sense of such expressions...

Perhaps, does it mean \begin{equation} \lim\limits_{g \to h} T\bigl( g \otimes h)= \lim\limits_{g \to h}\int_{\mathbb{R}^n \times \mathbb{R}^n } T(x,y)g(x)h(y)d^nxd^ny \end{equation} for each $h \in \mathcal{S}(\mathbb{R}^{n})$, where the limit $g \to h$ is in the Frechet topology of $\mathcal{S}(\mathbb{R}^{n})$?

If so, I see that \begin{equation} \lim\limits_{x \to y} \delta^n(x-y) \end{equation} must be characterized by the expression \begin{equation} \lim\limits_{g \to h}\int_{\mathbb{R}^n \times \mathbb{R}^n } \delta^n(x-y)g(x)h(y)d^nxd^ny = \lim\limits_{g \to h}\int_{\mathbb{R}^n } g(x)h(x)d^nx =\int_{\mathbb{R}^n } h^2(x)d^nx \end{equation} for each $h \in \mathcal{S}(\mathbb{R}^{n})$.

However, $h \to \int_{\mathbb{R}^n } h^2(x)d^nx$ does NOT define a linear functional..so I am confused..

multiplydistributions. What is your version of "Wick products", if you don't mind? $\endgroup$2more comments