Let $\mathcal{S}(\mathbb{R}^d)$ denote the $d$-dimensional Schwartz space. Evidently, the map $f\mapsto \bar{f}$, taking a function to its complex conjugate, is a continuous map of the Schwartz space to itself. Therefore, by the Schwartz kernel theorem, there exists a unique tempered distribution $K\in \mathcal{S}'(\mathbb{R}^d\times\mathbb{R}^d)$, such that $$<K,(f\otimes g)>_{\mathcal{S}'-\mathcal{S}} = \int_{\mathbb{R}^d} g(x) \overline{f(x)}dx,$$ where $<\cdot,\cdot>_{\mathcal{S}'-\mathcal{S}}$ is the distributional pairing and $(f\otimes g)(x,y) := f(x) g(y)$ is a function on $\mathbb{R}^{2d}$.
I am trying to give a direct proof of the existence and uniqueness of $K$ without on the kernel theorem, but I am unable to. Any suggestions?