Linear equations and bornology. (Linejnye uravneniya i bornologiya). (Russian) Zbl 0534.46004
Minsk: Izdatel’stvo Belorusskogo Gósudarstvennogo Universiteta im. V. I. Lenina. 200 p. R. 1.40 (1982).
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.
A fairly balanced presentation of parts of ”soft” and ”hard” analysis.
Reviewer: J.Lorenz
MSC:
46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |
47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |
46A08 | Barrelled spaces, bornological spaces |
46H30 | Functional calculus in topological algebras |
46E40 | Spaces of vector- and operator-valued functions |
46H05 | General theory of topological algebras |
47A60 | Functional calculus for linear operators |
47A10 | Spectrum, resolvent |
47D03 | Groups and semigroups of linear operators |
47E05 | General theory of ordinary differential operators |
34G10 | Linear differential equations in abstract spaces |
47F05 | General theory of partial differential operators |
47Gxx | Integral, integro-differential, and pseudodifferential operators |
45D05 | Volterra integral equations |