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The Schwartz kernel theorem, e.g, that all continuous linear maps from the space $\cal S(\mathbb R^m)$ of Schwartz functions on $\mathbb R^m$ to tempered distributions $\cal S'(\mathbb R^n)$ on $\mathbb R^n$ are given by "kernels" $K(,)$ in $\cal S'(\mathbb R^{m+n})$, is ${\rm Hom}(\cal S\otimes \cal S,\mathbb C)\approx {\rm Hom}(\cal S, \cal S')$, ...
The universal enveloping algebra functor $\mathcal{U}$ from Lie algebras to unital associative algebras is the left adjoint of the functor which assigns to each unital associative algebra a corresponding Lie algebra with bracket given by the commutator. This adjunction shows that the category of representations of a Lie algebra $\mathfrak{g}$ is in fact ...