This booklet consists of two parts: In the first a description of the theory of bornological vector spaces in the spirit of Waelbroeck and Hogbe-Nlend is given, following very closely the Lecture Note Math. 277 [Séminaire Banach (1972; Zbl 0239.46075)]. In the second half of the book the spectral theory of multiplicatively convex bornological algebras (Gelfand-theory, holomorphic functional calculus) is developed and applied to concrete problems of analysis: Volterra integral equations, ordinary and partial differential equations, Cauchy- and Goursat-problem. Here the equicontinuous bornology of the space of operators of a locally convex space is used systematically and the rôle of regular operators is emphasized (for such an operator A we have estimations of the type \(p(A^ nx)\leq M^ nq(x)\), p a given, q a seminorm to be found, \(M>0,n\in {\mathbb{N}})\). Roughly one can say, that for regular operators the classical Banach space theory goes through.
A fairly balanced presentation of parts of ”soft” and ”hard” analysis.