Schwartz kernel theorem in the case the distributions are induced by smooth functions..

Question

how can I show that if $A:C^\infty(\mathbb T^n)\rightarrow C^\infty(\mathbb T^n)$ is a continuous linear operators then there is a unique linear and continuous operator $K_A: C^\infty(\mathbb T^n\times \mathbb T^n)\rightarrow \mathbb C$ such that, $$\langle A\varphi, \psi\rangle=\langle K_A, \psi\otimes \varphi\rangle.$$ Remember that if $u\in C^\infty(\mathbb T^n)$ then $u$ induces a distribution by the pairing: $$\langle u, \phi\rangle=\int_{\mathbb T^n} u(x)\phi(x)\ dx.$$ Indeed, I don't know if this result holds, that would be a version of the Schwartz kernel theorem in the case the distributions are induced by smooth functions... Can anyone help me...