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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...

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It helps to know that $C^\infty(\mathbf{T}^n)$ can be equivalently topologized by the Sobolev norms $\| \cdot \|_{H^m}$ where $$\|f\|_{H^m}^2 := \sum_{\xi \in \mathbf{Z}^n} (1 + |\xi|)^{2m} |\hat{f}(\xi)|^2,$$ and $\hat{f}(\xi) = \int_{\mathbf{T}^n} e^{-i\xi x} f(x) \, dx$ are the Fourier coefficients of $f$. Indeed, for any $k \ge 0$ and for any integer $\ell$ strictly greater than $n/2$, one has $$ \|f\|_{C^k} \lesssim_k \|f\|_{H^{k + \ell}} $$ and $$ \|f\|_{H^k} \lesssim_k \|f\|_{C^{k + \ell} }. $$ In particular, $|\langle Af, g \rangle|\lesssim \|f\|_{H^n} \|g\|_{H^n}$ (say) for all $f, g \in C^\infty$.

Write $\mathbf{e}_\xi (x) = e^{i \xi x}$. Any $h \in C^\infty(\mathbf{T}^n \times \mathbf{T}^n)$ has a unique Fourier series representation $$h = \sum_{j, k} \langle h, \mathbf{e}_j \otimes \mathbf{e}_k \rangle \mathbf{e}_j \otimes \mathbf{e}_k;$$ by the equivalence of norms, this series converges in $C^\infty$. Thus $K_A$ must have the form $$ K_A(h) = \sum_{j, k} \langle h, \mathbf{e}_j \otimes \mathbf{e}_k \rangle \langle A \mathbf{e}_j, \mathbf{e}_k \rangle,$$ and this is in fact the desired $K_A$.

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