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According to the Schwartz Kernel Theorem and its variants, there are the canonical isomorphisms $$\mathcal{D}^{\prime} \left(X\right)\tilde\otimes \mathcal{D}^{\prime} \left(Y\right)\simeq\mathcal{D}^{\prime}\left(X\times Y\right),$$ $$\mathcal{E}^{\prime} \left(X\right)\tilde\otimes \mathcal{E}^{\prime} \left(Y\right)\simeq\mathcal{E}^{\prime} \left(X\times Y\right),$$ $$\mathcal{S}^{\prime} \left(\mathbb R^n\right)\tilde\otimes \mathcal{S}^{\prime} \left(\mathbb R^m\right)\simeq\mathcal{S}^{\prime} \left(\mathbb R^{n+m}\right),$$ where $E\tilde\otimes F$ is the completion of the space $E\otimes F$.
Roughly speaking, this follows from the fact that the corresponding spaces of test functions $\mathcal{D}$, $\mathcal{C}^{\infty}_c$, \mathcal{C}^{\infty}$, and$\mathcal{S}$are nuclear Fréchet spaces, and one has the canonical isomorphisms $$E^{\prime}\tilde\otimes F^{\prime}\simeq \left(E\tilde\otimes F\right)^{\prime}\simeq B(E,F),$$ when L(E; F'),$$provided that E and F are nuclear Fréchet spaces. (Here the duals carry the strong dual topology and the space B(E,F) L(E;F ') of continuous linear mappings is endowed with the topology of uniform convergence on the products of bounded sets.convergence.) As Johannes mentioned in his comment, a detailed presentation of the Schwartz Kernel Theorem and its versions for various spaces of distributions can be found in Topological Vector Spaces, Distributions and Kernels by Trèves. (More specifically, take a look at Chapt. 51, "Examples of Nuclear Spaces. The Kernels Theorem".) 2 added 490 characters in body; added 2 characters in body; deleted 2 characters in body According to the Kernels Theorem due to Laurent Schwartz (Kernel Theorem and its variants)variants, there are the canonical isomorphisms$$\mathcal{D}^{\prime} \left(X\right)\tilde\otimes \mathcal{D}^{\prime} \left(Y\right)\simeq\mathcal{D}^{\prime}\left(X\times Y\right),\mathcal{E}^{\prime} \left(X\right)\tilde\otimes \mathcal{E}^{\prime} \left(Y\right)\simeq\mathcal{E}^{\prime} \left(X\times Y\right),\mathcal{S}^{\prime} \left(\mathbb R^n\right)\tilde\otimes \mathcal{S}^{\prime} \left(\mathbb R^m\right)\simeq\mathcal{S}^{\prime} \left(\mathbb R^{n+m}\right),$$where E\tilde\otimes F is the completion of the space E\otimes F. Roughly speaking, this follows from the fact that the corresponding spaces of test functions \mathcal{D}, \mathcal{C}^{\infty}_c, and \mathcal{S} are nuclear Fréchet spaces, and there are one has the canonical isomorphisms$$E^{\prime}\tilde\otimes F^{\prime}\simeq \left(E\tilde\otimes F\right)^{\prime}\simeq B(E,F),$$if when E and F are nuclear Fréchet spaces. (Here B(E,F) is endowed with the topology of uniform convergence on the products of bounded sets.) As Johannes mentioned in his comment, a detailed presentation of the Kernel Theorem and its versions can be found in Topological Vector Spaces, Distributions and Kernels by Trèves. (More specifically, take a look at Chapt. 51, "Examples of Nuclear Spaces. The Kernels Theorem".) 1 According to the Kernels Theorem due to Laurent Schwartz (and its variants), there are the canonical isomorphisms$$\mathcal{D}^{\prime} \left(X\right)\tilde\otimes \mathcal{D}^{\prime} \left(Y\right)\simeq\mathcal{D}^{\prime}\left(X\times Y\right),\mathcal{E}^{\prime} \left(X\right)\tilde\otimes \mathcal{E}^{\prime} \left(Y\right)\simeq\mathcal{E}^{\prime} \left(X\times Y\right),\mathcal{S}^{\prime} \left(\mathbb R^n\right)\tilde\otimes \mathcal{S}^{\prime} \left(\mathbb R^m\right)\simeq\mathcal{S}^{\prime} \left(\mathbb R^{n+m}\right),$$where E\tilde\otimes F the completion of the space E\otimes F. Roughly speaking, this follows from the fact that the corresponding spaces of test functions \mathcal{D}, \mathcal{C}^{\infty}_c, and \mathcal{S} are nuclear Fréchet spaces, and there are the canonical isomorphisms$$E^{\prime}\tilde\otimes F^{\prime}\simeq \left(E\tilde\otimes F\right)^{\prime}\simeq B(E,F),$$if$E$and$F$are nuclear Fréchet spaces. (Here$B(E,F)\$ is endowed with the topology of uniform convergence on the products of bounded sets.)