# Convergence and Tempered Distributions

Let $$V = V(x,y)$$ be defined on $$\mathbb{R}^{n}\times \mathbb{R}^{n}$$ and suppose it is continuous in both variables $$x$$ and $$y$$ and satisfies $$\sup_{x,y\in \mathbb{R}^{n}}|V(x,y)| \le K$$ for some constant $$K$$. Now, define a bilinear map $$B: \mathcal{S}(\mathbb{R}^{n})\times \mathcal{S}(\mathbb{R}^{n}) \to \mathbb{C}$$ by: $$B(f,g) := \langle f, Vg \rangle \equiv \int dx dy f(x) V(x,y)g(y)$$ Once $$\mathcal{S}(\mathbb{R}^{n}) \subset \mathcal{S}'(\mathbb{R}^{n})$$ densely, there exits a sequence $$\{f_{n}\}_{n\in \mathbb{N}}$$ in $$\mathcal{S}(\mathbb{R}^{n})$$ such that $$f_{n}\to \delta_{x}$$ in $$\mathcal{S}'(\mathbb{R}^{n})$$, where $$\delta_{x}$$ is the Dirac delta distributin at $$x \in \mathbb{R}^{n}$$.

Now, if $$f_{n}\to \delta_{x_{0}}$$ and $$g_{n}\to \delta_{y_{0}}$$ in $$\mathcal{S}'(\mathbb{R}^{n})$$, does it follow that $$B(f_{n},g_{n}) \to V(x_{0},y_{0})$$?

• No, you need that $f_n\to \delta_{x_0}$ in the dual of $C^0\cap L^\infty$, or that $B$ is given by some Schwartz functions ie. is continuous on $S' \times S'$ – reuns Dec 18 '19 at 16:11
• You mean that $\int f_{n}(x)\phi(x)dx \to \phi(x_{0})$ for every bounded continuous function $\phi$? – IamWill Dec 18 '19 at 16:20