  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).
