Models of Linear Logic based on the Schwartz $\varepsilon$-product

From the interpretation of Linear Logic multiplicative disjunction as the $\varepsilon$-product defined by Laurent Schwartz, we construct several models of Differential Linear Logic based on usual mathematical notions of smooth maps. This improves on previous results, by R. Blute, T. Ehrhard and C. Tasson, based on convenient smoothness where only intuitionist models were built. We isolate a completeness condition, called k-quasi-completeness, and an associated notion stable by duality called k-reflexivity, allowing for a $$-autonomous category of k-reflexive spaces in which the dual of the tensor product is the reflexive version of the $\varepsilon$ product. We adapt Meise's definition of Smooth maps into a first model of Differential Linear Logic, made of k-reflexive spaces. We also build two new models of Linear Logic with conveniently smooth maps, on categories made respectively of Mackey-complete Schwartz spaces and Mackey-complete Nuclear Spaces (with extra reflexivity conditions). Varying slightly the notion of smoothness, one also recovers models of DiLL on the same $$-autonomous categories. Throughout the article, we work within the setting of Dialogue categories where the tensor product is exactly the $\varepsilon$-product (without reflexivization).