Given is a continuous linear functional $T:C_c^0(\mathbb{R})\otimes C_c^0(\mathbb{R}) \rightarrow \mathbb{R}$ where $C_c^0(\mathbb{R})$ is the space of continuos functions with compact support. Since $C_c^0(\mathbb{R})\otimes C_c^0(\mathbb{R}) \cong C_c^0(\mathbb{R}^2)$ via the Riesz representation theorem i can find a Radon measure $\nu$ on $\mathbb{R}^2$ such that
$$T(\psi)=\int_{\mathbb{R}^2}\psi(x)\nu(dx) \; \; \; \forall \psi \in C_c^0(\mathbb{R}^2) $$
where here $T$ is understood to be a continuous linear functional from $C_c^0(\mathbb{R}^2)$ to $\mathbb{R}$.
Is it true then that for $T:C_c^0(\mathbb{R})\otimes C_c^0(\mathbb{R}) \rightarrow \mathbb{R}$ there exists a Radon measure $\mu$ on $\mathbb{R}$ such that
$$T(\phi_1 \otimes \phi_2)=\int_\mathbb{R}\phi_1(\xi)\phi_2(\xi)\mu(d\xi) \; \; \; \forall \phi_1,\phi_2 \in C_c^0(\mathbb{R}) ?$$
Best Answer
To complement the answer above: There are several subtleties that you are missing.