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Axioms for convenient calculus. (English) Zbl 1076.46031

Let \(\Gamma\) be a class of structured real vector spaces, and for every \(E,F \in \Gamma\) let \({\mathcal S}(E,F)\) be a set of maps \(f : E \to F\), called smooth. If \(E^*\) is the algebraic dual of \(E\), \(E'\) denotes the set \({\mathcal S} \cap E^*\). The author sets three simple axioms on the sets \({\mathcal S}(E,F)\) which imply that (i) \(\Gamma\) and the smooth maps form a category, (ii) smooth maps are completely determined by the sets \(E'\), (iii) \((E,E')\) is a dualized vector space. Every dualized vector space \((E,E')\) defines a bornological vector space \((E, {\mathcal B}_E)\) and vice versa. It is proven that a linear map \(f : E \to F\) is smooth (belongs to \({\mathcal S}(E,F)\)) iff it is a morphism of dualized vector spaces, or, equivalently, iff it is a morphism of bornological vector spaces. Thus, \({\mathcal S}(E,F)\) is recovered if one knows either the duals \(E'\) and \(F'\) or the bornologies \({\mathcal B}_E\) and \({\mathcal B}_F\).
Next, it is proven that the third axiom holds for a vector space \(E\), if and only if \((E, {\mathcal B}_E)\) is separated and Mackey complete. Thus, the author approaches the notion of convenience via bornology: a convenient vector space is a bornological vector space which is linearly generated, separated and Mackey complete. Hence, we arrive at the main result of the present work: the axioms set here hold for the category Con\(^{\infty}\) of convenient vector spaces and smooth maps (in the sense of the convenient setting). They also hold for every full subcategory of Con\(^{\infty}\) containing \({\mathbb R}\) and, conversely, every model for the axioms is equivalent to a full subcategory of Con\(^{\infty}\) containing \({\mathbb R}\). Thus, Con\(^{\infty}\) is a maximal model of the present axiom system.

MSC:

46G05 Derivatives of functions in infinite-dimensional spaces
46A17 Bornologies and related structures; Mackey convergence, etc.
46M15 Categories, functors in functional analysis
18B99 Special categories
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References:

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