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 $B(f_{n},g_{n}) \to V(x_{0},y_{0})$?

  • $\begingroup$ 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 $S' \times S'$ $\endgroup$ – reuns Dec 18 '19 at 16:11
  • $\begingroup$ You mean that $\int f_{n}(x)\phi(x)dx \to \phi(x_{0})$ for every bounded continuous function $\phi$? $\endgroup$ – IamWill Dec 18 '19 at 16:20

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