Linear spaces and differentiation theory. (English) Zbl 0657.46034

Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs and Tracts. Chichester (UK) etc.: Wiley (ISBN 0-471-91786-9). xvi, 246 p. (1988).
A dualized vector space is a vector space (over reals) together with a given subspace of its algebraic dual. DVS is the category of dualized vector spaces (the morphisms are the linear maps \(m:E_ 1\to E_ 2\) satisfying \(m^*(E_ 2')\subseteq E_ 1')\), CBS is the category of convex bornological spaces (with linear bornological maps as morphisms), \(\delta\) :CBS\(\to DVS\) the duality functor and \(\sigma_ b:DVS\to CBS\) its right adjoint. A preconvenient vector space is a dualized vector space invariant under the functor \(\delta \circ \sigma_ b\); a convenient vector space is a preconvenient vector space which is separated and whose Mackey convergence structure is complete. The category of convenient vector spaces Con has proved to be the “convenient” category of vector spaces for the differentiation theory in infinite-dimensional spaces. Note that Con contains all Fréchet spaces. The feature of the differential calculus in Con is the fact that k-fold differentiability (with a Lipschitz condition on the derivatives) for maps can be reduced to that of (real) functions of one real variable.
The book will be of interest to “pure” mathematician as well as to theoretical physicists.
Reviewer: J.Danes


46G05 Derivatives of functions in infinite-dimensional spaces
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
46M15 Categories, functors in functional analysis